Co l u m b i a Sc i e n c e Re v i e w Vol. 5, Issue 2: Spring 2009

Mirror Neurons Monkey-See, Monkey-Do Magnificent Desolation? The Search for Extraterrestrial Life Squaring the Circle The Ancient Delian Problem Martin Chalfie An Interview with the Nobel Prize Winner

Spring 2008 1 Columbia Science Review

Editorial Board

The Editorial Board biannually publishes Columbia Science Review, a peer-reviewed science publication featuring articles written by Columbia students.

Editor-in-Chief Jonathan Mo

Content Editor Content Reviewer Layout Editor Sandeep Soman Pradeep Bandaru Ying Li

Board of Editors & Reviewers Layout Saroja Bangaru Elizabeth Robinson Ivy Chen Sweta Agrawal Elizabeth Robinson Ivy Chen Abigail Wilpers Alanna Vaughns Danielle Bitterman

Administrative Board

The Executive Board represents the Columbia Science Review as an ABC-recognized Category B student organization in Columbia University.

Niccola Pérez, President Chris Schell, VP of Public Relations Clay Wiske, VP of Finance and Budgeting Shelly Zhu, Managing Editor Laika Simeon, Managing Editor Roland Zemla, Chair of Recruitment Christine Yeh, Secretary

The Columbia Science Review

The Columbia Science Review strives to increase knowledge and awareness of science and technology in the Columbia community by presenting engaging and informative approaches to contemporary science and technology that include, but are not limited to:

• Exploration into contemporary issues of science, including research, policy, and opinion • Features on current faculty • Opportunity for students to publish their scientific research

Special Thanks to:

Student Development and Activities Office of Undergraduate Admissions David Milch Karen Sagall Rebecca Bromberg

2 Columbia Science Review Co l u m b i a Sc i e n c e Re v i e w

04 Letters 05 Cocktail Science 06 Bioethics Newsletter 09 Magnificent Desolation 12 Evolutionary Genetics 16 Chatting with Chalfie 19 The Wonderful World of Sleep 21 Blue Brain 22 Tower of Hanoi 24 The Curious Case of the Beans 27 The Delian Problem 30 Mirror Neurons

12 16 19 24

Evolutionary Genetics Chatting with Chalfie The Wonderful World of The Curious Case of the Sleep Beans: G6PDD

COVER PICTURE: Using genes from various organisms, scientists at Harvard University have developed a technique to light up brain cells in mice. These brain cells are from the dentate gyrus, a part of the brain associated with memory, and each one randomly ex- presses fluorescent proteins of different colors. Credit: Livet and Lichtman, Harvard University.

Spring 2009 3 Le t t e rs

Letters

Despite nearly 400 years of success, science still faces tremendous challenges in obtaining the popular support of many. A key problem of science is that it has always seemed detached from the human condition. Unlike enduringly popular religion, which has promised everything from eternal life to salvation for our sins, science is often viewed as a cold, unfeeling stance towards the universe; a concept lacking humanity, and thus, enthusiastic adherents. But this is not true.

Science lies at the very core of what makes us human. Let us call it Humanism instead. Humanists remind us that the human condition is universal, and most importantly, that it can be improved through our mutual cooperation and the ability of the human mind through the scientific inquiry of nature.

Science is one of the greatest forces for good in the world that we have ever discovered. But I believe this is obscured because we miss the Humanism that underlies it. It is the most basal human faith, one that recognizes our biological commonality, one that celebrates the awesome beauty and mystery of our universe, and one whose objective is the advancement of our collective condition without exclusion. Its central tenant is faith in the human ability to understand, to question, to wonder, and to use that great wonder as a source of inspiration, that we might find, however elusive, the truths of our existence, and the peace of our mortality. Personally, I find this the most reassuring and altogether most fulfilling of endeavors, which I hope in all its majesty calms even the most restless of spirits, even the most parochial of critics.

Science is for everyone; it is a shared part of our human experience. At the Columbia Science Review, we hope this common bond, this ‘voice of life,’ calls you to come and learn.

-Jonathan L. Mo, Editor-in-Chief

As with every issue of the Columbia Science Review, we strive to explore…well… science! But what is science? Commonly defined as an objective study of the physical world, the human race has developed a method—the scientific method—for regulating this investigative process. The great success of this method in allowing us to make conclusions about the environment we inhabit has led us to neglect many subjective discussions of science, rendering countless studies, ideas and proposals invalid. So what do we lose in our often blind pursuit of objectivity, if anything at all? Is there a place for rock music in formulating new genetic sequencing techniques? Could sculpture add to our of quantum theory? Does poetry allow further conceptualization of an intricate computer code? Conversely, could knowledge of a plant cell’s components inspire acrylic painting? Could identification of a novel animal species in the Amazon give rise to a new movie script?

It is with these questions that I would like to dedicate this issue to you, the reader, so that you may expand your frame of reference with respect to whichever discipline you have chosen for study. Academic disciplines have never been more interconnected, and they continue to meld and mesh every day, creating new and exciting ideas in the process. The perpetual formation of these junctures is redefining the word science and I call upon you to use this issue of CSR as a platform for further exploration of this interdisciplinary, amorphous, and limitless word.

-Niccola Pérez, President

4 Columbia Science Review Cocktail Science Cocktail Science

elieve it or not, animals can rain from the sky. This relatively common meteorological phenomenon has been occur- Bring in Honduras for more than a century on a yearly basis between the months of May and July. Live fish literally rain down from the sky during a violent thunderstorm accompanied by strong winds and heavy rain. Although there is no valid scientific explanation for this bizarre phenomenon, many locals believe that Father Jose Manuel Subirana, a Spanish Catholic missionary, rains the fish down upon Honduras to provide food for the poor and malnourished. Can you think of a better (and cooler) way to go fishing? -Pradeep Bandaru

esearchers at the UC San Diego School of Medicine have iden- Rtified a new possible link between cancer and consumption of red meat and milk products. As a result of consuming red meat, humans produce a type of sugar molecule called Neu5Gc, which tissues can incorporate; anti-Neu5Gc antibodies are then produced as an immune response, potentially leading to chronic inflammation and heightened cancer risk. Tumor tissues have been shown by these researchers to contain a disproportionate amount of Neu5Gc compared to healthy, normal human tissues. -Christine Yeh

he biodiversity of our planet, Tusually quantified by the number of species, is declining considerably, to a point where biologists have characterized this current time period as a mass extinction event. Coincidence? Not likely. Many ecosystems are responsible for properly handling carbon dioxide, low biodiversitytranslates into less carbon dioxide control. Simply put, the loss of species on our planet – corals and vegetationespecially – is one of the largest contributors to global warming patterns. But it is the most overlooked by the general populace and the most affected by human actions. We derive many benefits from increased biodiversity, such as decreases in pathogens, increased food productivity, and cleaner water. The New York tap water is safe to drink solely because the Catskill Mountains and healthy vegetation in the area work as an effective irrigation system to filter the harmful chemicals out of the water. -Chris Schell

Nippostrongylus brasiliensis (parasitic worm) (40x) Courtesy of Nikon Small World

Spring 2009 5 Bioethics Newsletter

A Bioethical Approach to Gene Therapy and its Role in the Future of Mankind

Introduction

Marlee Ickowicz and Larry Geyman he field of gene therapy has under- to have any effect. In addition, the body Tgone drastic changes in the last de- could attack the virus vector and tag it as cade, growing from a nascent field to a a foreign substance. The original virus also burgeoning science. Gene therapy seeks might become active and infect the recipi- to cure diseases by fixing problems at ent. Although gene therapy is extremely their source: genes. The therapy consists promising, advancements are slow. of inserting a “healthy” gene into the ge- The ethical considerations of gene nome of a patient to replace an “abnor- therapy have been debated in political, mal” gene. This process is typically carried religious, and scientific circles since 1967. out by an altered virus, called a vector, Initially, the controversies presented by that is able to deliver human genes into each of the aforementioned parties were cells. The two types of gene therapy are reflected by lackluster legislation that germ-line gene therapy and somatic gene failed to come to a consensus on how therapy. In germ-line gene therapy, viruses to deal with the issue. In October 1999, are used in reproductive cells to alter the the death of 18-year-old Jesse Gelsinger genetic composition of preembryos and marked the first reported fatality caused effectively eliminate their diseased genes. by gene therapy. His death, which was fol- They can also be used in the reproductive lowed by six other gene-therapy related cells of adults, eliminating the possibility deaths, complicated the ethical dilemma. that their offspring could inherit a disease. In the years that followed, significant ad- Somatic gene therapy, on the other hand, vancements in gene therapy techniques is less invasive, expensive, and controver- allowed scientists to cure deafness and sial because it targets genes in adults’ inherited blindness and re-engineer im- non-reproductive cells. mune cells to attack cancer cells. These The current status of gene ther- extreme positive and negative reports apy is both promising and disheartening. heightened the ethical debate surrounding Although there have been outstanding gene therapy and pressured political and improvements in technique and applica- ethical pundits to play more decisive roles tion, there are still numerous procedural in future research and practice. Currently, difficulties. Because of cell death and cell the debate about somatic gene therapy division, there is no guarantee that the has been tabled, but debate regarding cells into which the genes are inserted germ-line gene therapy continues. will survive long enough for the therapy

6 Columbia Science Review Bioethics Newsletter

if one of the deaths is a future Nobel Peace Prize winner, an accomplished international diplomat, or even your average Fo r t h e Sa k e o f Ou r Sp e c i e s : “Joe the Plumber”? Is any life worth risking for uncertainty, especially when the outcome could possibly be harmful for An a r g u m e n t i n o p p o s i t i o n our species? Germ-line gene therapy is currently available only t o Ge r m -l i n e g e n e t h e r a p y to people who can afford the pricey procedure. If this trend continues, affluent people will have the privilege of altering Marlee Ickowicz their genes or their children’s genes, while the masses will not. This inequity based upon financial stratification has the umans have certain essential characteristics that make potential to increase the gap between the wealthy and the them distinct from other species. Although gorillas can H poor. Although the possibility of a healthier and wealthier learn sign language and parrots can repeat sentences, only species is appealing, it would have negative consequences humans can reason, abstract, form complex language, and because it disrupts the natural order of intermixing and di- display a wide range of emotions. These qualities are fun- versity. damentally genetic. There is an organic process of genetic Finally, germ-line gene therapy has the potential to accel- mutation, selection, and permutation that makes the hu- erate the demise of the human species. If scientists com- man species unique. In other words, our genes make us hu- pletely eradicated disease, humans would overpopulate at man. What if we alter them? With gene therapy, we would an unnaturally high rate, making the fight over earth’s al- have the potential to become healthier and more normal. ready limited resources a losing battle. Other possible haz- However, we would also have the power to alter what is ards range from bioterrorism to misguided regulation. If gene essentially human. therapy technology gets into the wrong hands, the ability Genes are passed on from generation to generation. to manipulate genes could be used as a weapon. Moreover, They give our grandchildren ten fingers, ten toes, and all of if scientists and researchers manipulated their own genes, the other elements necessary to perpetuate the species. could they be trusted to regulate ours? If scientists alter these fundamental units of inheritance, A return to the financial analogy seems appropri- humans have the potential to pass down traits that are ar- ate. Amidst global economic downturn and fear of reces- tificial or synthetic. Manipulating this process presents the sion, lawmakers, businessmen, and politicians have scram- possibility of creating a new species and perhaps perma- bled around Wall Street and Main Street to find a remedy nently impairing our own. that will hold the fragile economy together. Now imagine Consider the following analogy regarding the cur- scientists frantically struggling to salvage our species after rent state of our financial markets. The massive financial the onslaught of gene therapy’s potentially hazardous re- failure we are experiencing is a result of artificial and syn- percussions. Time is of the essence and, unlike the financial thetic financial instruments that were created without an crisis, we do not have the luxury of trial-and-error solutions understanding of the potential consequences. Political and because the survival of the human species is at stake. If we economic experts have still not agreed on a solution for could go back in time and stop the financial crisis from hap- the broken financial system. Similarly, we have not seriously pening, would we? I would hope so. considered the consequences of germ-line gene therapy and It seems clear that the risks are too great and the ben- we do not have a solution to mend a potentially “broken” efits are too small. Gene manipulation would alter our spe- human species. cies, renouncing what is essentially human and accelerating Although gene therapy currently manipulates genes that human demise. Tampering with genes–humanity’s essential carry particular strains of disease, scientists have the po- building blocks and the fundamental units of inheritance– tential to alter all diseased genes, and furthermore, to alter threatens human naturalness and calls into question the genes and human traits not associated with disease. Who future of our species. draws the line?

On a practical level, scientists and researchers have Refrences found that there are uncertainties with germ-line gene ther- apy, as manifested in the deaths of at least seven people. (March, 1994). Scope Note 24: Human Gene Therapy . National Reference Center for Bioethics Literature, Retrieved 10/19/2008, from http://bioethics. In addition, researchers have reported unanticipated conse- georgetown.edu/publications/scopenotes/sn24.htm quences such as induced leukemia (Kaiser, 2003). A utilitarian Blackmore, Susan (2004). Consciousness: An introduction. New York, New York: approach may consider this justifiable because thousands Oxford University Press. Caplan, Arthur (2002). If Gene Therapy Is the Cure, What is the Disease?. In of people may benefit from germ-line gene therapy. How- The American Journal of Bioethics [Web]. Retrieved 10/29/2008, ever, because the outcome of gene therapy is unclear, ad- from http://www.bioethics.net/articles.php?viewCat=6&articleId=58 Kaiser, Jocelyn (2004, January 24). Gene therapy: Seeking the cause of induced ditional deaths are likely to occur during the process of de- leukemias in X-SCID trial . Science, Vol. 299. no. 5606, 495. veloping effective and safe gene therapy procedures. What

Spring 2009 7 Bioethics Newsletter

Ma s t e r i n g t h e Pr o c e ss o f Ev o l u t i o n : An Ap p e a l t o In c r e a s e Ge n e Th e r a p y Re s e a r c h

Larry Geyman ust like any scientific field, gene therapy will grow and the molecules that define us. If the new, improved gene has Jdevelop until one day it becomes practical enough to but one incorrect nucleotide, the person may die. However, be applied to humans. But what are the justifications for this should not diminish the potential benefits of improved allowing gene therapy to develop to such a level? Gene general health, increased longevity, and the eradication of therapy is, no matter how you look at it, tampering with dozens of fatal diseases. In a world in which cancer causes nature and with DNA, and it will have huge implications in 7.1 million deaths annually (Gene Therapy 2008), and dis- the future. Nonetheless, I claim that gene therapy research eases run rampant throughout developing countries, gene should continue at its current pace and I welcome advance- therapy offers too many benefits to even consider halting ments in this exciting new field. its development. Some argue that gene therapy is tampering with na- Thirdly, we must look at future implications of ture. They suggest that by altering our genes, we would lose gene therapy, particularly its large-scale effects. If all dis- what is “natural” about us. However, to be human is to be eases were eradicated, the human population would sky- constantly changing, so there is no constant “human” entity rocket. However, because of our planet’s limited resources, in the first place. We are not a we would be headed for disas- static species at all. Genes are ter. However, if gene therapy “lost” throughout history and were developed in tandem with are retained in our genome as renewable energy technology, more-or-less useless pieces of the planet would be able to so called “junk” DNA. support more life. In addition, genes them- Evolution takes millions of selves are constantly changing. years to be visible, so try to Experiences shape us, but more think of gene therapy as “ac- correctly, they shape our DNA. celerated evolution.” It is no dif- Through external influences, ferent than traditional natural the body turns on certain genes selection-induced evolution ex- and turns off others, form- cept that now we have become ing what makes each individual intelligent enough that we are unique. Yet this “uniqueness” is able to select traits without not constant. Think of homo habilis, our ancestor. To these the aid of natural selection. “There is nothing holy about a ancient humans, to be human was to be a homo habilis. But fragment of DNA,” states David King, Director of Genetic to us, to be human is to be a homo sapien. What is deemed News (Bergeson, Ethics of Gene Therapy, The 2008). And human is subject to change. Altering genes synthetically is there is nothing unnatural about taking evolution into our not unnatural; rather, it is taking control of what nature has own hands and guiding it. We have become sufficiently intel- been performing for billions of years. ligent that we can evaluate the risks and plan ahead for the One might also suggest that the costs simply future to create a society in which all will be able to enjoy outweigh the benefits. However, every scientific discovery the miraculous benefits of gene therapy. comes with its share of risks. From a utilitarian perspective, one can clearly see that if gene therapy will save more indi- Refrences viduals than it will harm in the process of perfecting it, then Bergeson, Emilie R. “Ethics of Gene Therapy, The.” 1997. North Dakota State the research behind its development is worthwhile. How- University. 22 Oct. 2008 . “Gene Therapy.” 19 Sept. 2008. Oak Ridge National Laboratory. 22 Oct. 2008 higher than the risks for the development of vaccines for .

8 Columbia Science Review Astrobiology

“MAGNIFICENT DESOLATION?”

Astrobiology’s epic search for life out on the final frontier

Jonathan Mo

A Pale Blue Dot For all the magnificence of life on earth, we have hen Edwin “Buzz” Aldrin first set foot on the moon in long considered space to be a lifeless entity, absent evi- W1969, his words summed up the human experience of dence to the contrary. Science-fiction writers have long outer space: “Beautiful. Beautiful. Magnificent desolation.” made their living inciting the human imagination about aliens The stars have always held a unique place in the history and the possibility of life beyond the earth. The concept of mankind. Millenia ago our ancestors first described the of standing alone on what legendary astronomer Carl Sa- glory of the heavens, lands our race long ago dedicated to gan called our “pale blue dot” of a planet has fueled many the gods and the unknown. Try as we might, despite de- creative attempts to explain the phenomenon of extrater- cades of probes, observations, and incalculable theorizing, restrial life. the frontier beyond our tiny world ever remains a source of Yet in the last decade, we have begun to discov- nature’s most profound myster- er hundreds of extra-solar planets, or exoplanets: worlds ies. Looking up at a night sky of- around other stars. In addition ten fills young and old alike with Within the next 50 years, we may to leading a foundational study wonder, and might cause even very“ well witness the discovery of of life on earth, we draw nearer the most jaded astronomer to to truly searching for life in the ask the grandest of questions extra-terrestrial life. cosmos. In fact, we live in one of his unique laboratory: the uni- ” of the most exciting eras of sci- verse itself. entific exploration ever. By demystifying some aspects of outer space, we On 13 November 2008, two independent groups, have answered some of the greatest questions in human his- one using the Keck and Gemini ground telescopes (Marois et tory, from the ‘Copernican Revolution’ concept that humans al. 2008), and the other using the Hubble Space Telescope did not lie at the center of the universe to the realization (Kalas et al. 2008), announced the first optical and infrared that our home planet is not some customized construct “ir- photographs of extrasolar planets orbiting stars other than reducibly complex,” but rather a natural consequence of the our own. The first group imaged three exoplanets (Image laws of physics. Theology, philosophy, and society at large, 1) around the star HR8799, 125 light years from earth. The in addition to astronomy, all have a great stake in the study second group photographed a planet orbiting the nearby of the stars. And yet in the midst of this great confluence star Fomalhaut (Image 2), visible to the naked eye at 25 light of science, faith, and ever expanding humility, perhaps the years distant. These large gas-planets are known as ‘hot Ju- most profoundly human question has never been answered. piters’ because they share several evolutionary and physical Are we indeed alone in the universe? characteristics with our eponymous solar system neighbor,

Spring 2009 9 Astrobiology

and are still highly energetic, emitting enough light to make But what if, instead, life represents an inevitable and com- their photography possible. As coronagraphs (used to blot mon emergent property of chemistry? The seeming ther- out stellar light) improve in the next few years, the optical modynamic inequilibrium of life processes emerges from a photography of smaller exoplanets will become possible. foundation that absolutely follows the laws of physics. Bi- With these better imaging techniques and the discovery of ology, then, is not a separate science, but is rather a subject more worlds, the actual spectroscopic search for life can that occupies the final chapter in a physics or chemistry begin in earnest. text. It is the final result, the end product, of a universe Within the next 50 years, we may very well wit- which follows some very fundamental laws. ness the discovery of extraterrestrial life. The technology This is different from claiming that biology is based to seek for it exists, the methodologies and mathematical on the other basic sciences. It is instead a claim that the models are ready, and the scientific background to analyze universe actually produces life, as a product of sorts, and such a discovery is more or less complete. We stand on the that it can be decomposed into its requisite equations that brink of what might well be the most jarring discovery in govern its very statistical origin. It can be a profoundly al- the history of human existence. tering concept when taken in the context of theology and faith. What is Astrobiology? HR8799 ccording to astronomer Dr. Jona- In My Lifetime? Athan Lunine of the University of s of print, NASA’s Jet Propulsion Arizona, author of a popular text on ALaboratory reports that 322 ex- astrobiology, “Astrobiology is the mul- trasolar planets have been discovered tidisciplinary study of the origin, distri- in just a decade of limited searches. bution, and evolution (past and future) We have yet to find a ‘terrestrial’ type of life,” (Lunine 2005). planet like the Earth, but this is likely Astrobiology could be seen as due to their small size, not necessar- the ultimate level of abstraction of bi- ily their rarity. Despite heavy budget ology, the most encompassing view of cuts at the world’s space agencies, life as a whole. At its highest levels and with a likely reduction in the over- biology presumably reaches beyond all research and NASA budgets of the Image 1 (Credit: Christian Marois and Bruce the earth into the realm of astrono- Macintosh) United States, the prime leader of this my, leading the two sciences to find search, we are still poised to find earth- in one another a common ground–our like planets before the end of the next entire universe. The study of life on Earth then, the tradi- decade. tional study of biology, becomes the study of life in the Missions with names like Darwin, Kepler, and the universe as a whole: “astrobiology.” two-part Terrestrial Planet Finder have long been planned Of course, astrobiology encompasses so much and prepared for execution, but simple funding constraints more than just this grander view of life. It concerns itself have grounded them. As soon as these missions, and likely with the study of the earth and geology, meteorology, others, launch, we will discover how common such earth- geochemistry, climatology, and environmental and plan- like planets are, which will inform our search for life. etary sciences all find great relevance. Likewise, chemistry, Predictions vary widely, with some of the scientific physics, medicine, engineering, and of course all fields of bi- community siding with the aptly named “rare-Earth hypothe- ology and astrophysics are united in this great collusion of sis” that our planet’s ability to sustain life, while not unique, the sciences. is likely an uncommon occurrence. While of course Earth- But ultimately, due to its conglomerate nature, as- type planets are not the only worlds we might imagine life trobiology has a specific thesis. It seeks to explain the fun- to inhabit, they remain our starting point in a search that damental basis for the development of biochemistry. Is could well reach beyond the imagination. life a natural outcome of physical laws? That is to say, Others believe that Earth-type planets are plenti- does life emerge in the universe along a basis of rules and ful, and that we will find many hundreds of them nearby in fundamental principles, or does it represent some incredible our region of the galaxy. We will not know for sure until accident here on Earth? This might seem to draw an obvi- surveys of nearby stars are conducted. This is difficult to do ous answer from a chemist or a physicist: life is a product of for many reasons, including the incredibly small size of plan- basic rules of physics and chemistry. ets compared to their parent stars and the light of these But the claim can be seen as broader and more stars, which blinds our view. subtle. Biology has always occupied a special place in the Once even a single terrestrial world is found, the sciences, mainly because of its apparent uniqueness to Earth. burning question will be how to determine the presence of

10 Columbia Science Review Astrobiology life on this planet. Life fundamentally alters the character- wonderment, with vibrancy, and with life. istics of its host planet, and thus produces various signals, Astrobiology, perhaps more than any other field called biosignatures, that indicate its existence. Still, they in science, may yet bring us the final Copernican Revolu- are not easy to read, and a subspecialty has emerged in the tion – the realization that life, like our position in the cos- field of astrobiology dealing with biosignatures and how to mos, and the uniqueness of our Earth, is not some great detect and analyze them. conceit of a divine creator, or While this is certainly still an abstraction too complex to an underdeveloped study, it reconcile, but is rather an inevi- is rapidly accelerating in an- table consequence that lies at ticipation of the impending the very heart of physics and discovery of terrestrial plan- chemistry; as stars and planets ets. are created by gravity, so too Biosignatures in- does life represent a funda- clude most of the spectral mental outcome of the laws features of a planet that of the universe. we can obtain from the We would finally say of light we receive from it. A the cosmos, not even a cen- planet covered in vegetation tury after we first walked on of some kind, even incred- the moon, that while magnifi- ibly bizarre by Earth stan- cent, they are not a desolate dards, would produce what wasteland consigned only to it is called a vegetative-red- the study of astronomy, but edge. The reflection of light Image 2 (Credit: NASA, ESA, and Z. Levay (STScI)) are instead a living system of by surface flora would emit which we are merely specimens a pronounced signature in on both sides of the lens. the infrared spectrum, a particular reflection characteristic References of pigments used in photosynthesis. Kalas, P., et al. Science. 13 November 2008 (10.1126/science.1166609). There are more basic biosignatures, including the Lunine, Jonathan I. Astrobiology. Pearson Education, Addison Wesley Publish presence of ozone, oxygen, and water. Even with the dis- ers. San Francisco, 2005. Marois, C., et al. Science. 13 November 2008 (10.1126/science.1166585). covery of ‘another Earth,’ the confirmation of life on the NASA Jet Propulsion Laboratory. PlanetQuest: Exoplanet Exploration: planet could remain contentious, as it might be difficult to http://planetquest.jpl.nasa.gov/. Accessed 14 November 2008. make conclusive observations. Life does not announce its presence so readily, and an entirely new field of science will likely emerge dedicated solely to the study of this one, or perhaps many, planets. But in the midst of this scientific cataclysm might well appear evidence that we should not, as the history of science instructs, find surprising. In my lifetime? Perhaps. We live in the first era during which the technology to even ask these greatest of questions exists. The mission of searching for extraterrestri- al life is one that many scientists have considered for much of history, but could never accomplish. It is finally time to make good on these ancient questions and search for answers. If life is common in the universe, or even if we get lucky and find a planet nearby that supports it, it might not be 15 years before every history or science text, and indeed nearly every human conception of our place in the universe and the course of our future, is rewritten forever. It could well be longer. But it is no longer an ex- tant abstraction or solely a theoretical question for sci-fi authors. We are exploring at last, looking for evidence of something many scientists, at least at some point, have per- haps wanted to believe: that the universe is not some deso- late void that we call “outer space,” but is rather filled with

Spring 2009 11 Evolutionary Genetics Evolutionary Genetics

Chris Hanks pproximately six million years ago, a population of ap- become more common in the next generation. Aproximately 50,000 to 100,000 primates in Africa split So, where do new genes come from? Every sperma- into two groups for unknown reasons. Breeding between the tozoa or egg that an individual produces contains a copy of groups, and thus the transfer of genetic information, became half of their genetic code, in the form of DNA, which will less common. With this uncoupling, these two populations eventually go to contribute to the genetics of their off- began to change and differentiate, as each was subjected to spring. The cellular machinery responsible for copying this separate environmental factors. This was an event of spe- DNA makes mistakes - one “letter” of the code mistakenly ciation, the process by which one species splits into two. replaces another - creating small and novel changes in the Although it was an unusually complicated and prolonged genes of their offspring. These are known as “point muta- example of the phenomenon, with the two groups recon- tions,” and as they affect the genetic code and therefore the necting and hybridizing before splitting again, ultimately, the biology of the offspring, they will be acted upon by natural differentiation became so pronounced that the split was selection – if a change is harmful (as the vast majority are), made complete and permanent. One species of primate had the offspring will be less likely to pass it to the next genera- become two. tion, while a beneficial change is more likely to be passed One of these groups would go on to split again lat- on, increasing the prevalence of that changed gene over er, and become what we know today as the Common and time. Many mutations, however, do nothing – they change Pygmy Chimpanzees (Pan troglodytes the sequence of a gene, but not the and Pan paniscus, respectively). The actual function of that gene. These other would become human beings are “silent mutations,” By contrast, a (Homo sapiens). mutation that does affect the func- Historically, this puzzle has tion of a gene, and can be selected been approached through interpre- for or against, is called a “non-silent tation of the fossil record, but the mutation.” great advances in recent genetics re- Back to our common ancestor, the search, including the release of the primates in Africa six million years sequenced human genome in 2003, ago. They had their own genome, and that of the common chimpan- with their own sets of genes, very zee in 2005, have given us a new similar to ours. Because every indi- looking glass through which we may vidual gene from our common an- study our origins. Accordingly, many cestor split in two when our species new techniques in analytical genet- did, the individual genes that make ics have been developed for the up contemporary humans and Chim- study of evolutionary biology. panzees are very similar - they’ve all changed in the past six million years, Ev o l u t i o n a n d Mu t a t i o n s but they each come from a common hen we say that the pre-humans and pre-chimpanzees template. Genes that share their own “common ancestors” Wdifferentiated over time, what we really mean is that are called “homologous genes,” and by analyzing the muta- the overall genetic makeup of each group began to change tions that they have collected since our divergence from independently, and there are many factors that can cause Chimpanzees, we can learn quite a bit about our evolution- this, including natural selection and genetic drift, wherein ary history. genes that are less relevant to survival will become more or less common simply by chance. For example, in a given Mo l e c u l a r Cl o c k s generation, if blue-eyed individuals happen to have a greater et’s look first at silent mutations. Since evolution cannot number of children than brown-eyed people, blue eyes will Lact upon them, they are only subject to genetic drift - 12 Columbia Science Review Evolutionary Genetics they appear in an individual, start to spread around a pop- of time since two species diverged by analyzing total diver- ulation, and become more or less common due to simple gence, we can compare the times of divergence of a vari- chance. As time progresses, one of two things will eventu- ety of individual genes to estimate the population size of a ally happen to a silent mutation: it will either become less common ancestor at speciation. and less common and die out, or it will become more and First, picture the simple case, that there is a species more common until it’s found in every member of the spe- that is very small in number, and that undergoes a speciation cies. A mutation that does this, and reaches a prevalence event. Because it is easier for mutations and gene types to of 100%, is called “fixed” and becomes a permanent part of reach fixation, or to die out, in a smaller population, there the genome of that species. As time passes, more and more is not very much genetic diversity in this group - genes have mutations will become fixed in a species. comparatively few variants, and not very many mutations This means that we can look at homologous genes reacting to genetic drift are present. After the speciation between two or more species, count up the mutations that event, the genetic makeup of each group will start to change differentiate them, and determine which are silent and which and differentiate immediately, and when individual genes are are non-silent. The number of silent mutations present will analyzed using the Molecular Clock, they will all appear to correlate with how much time has passed since these spe- have started diverging at approximately the same time. cies diverged from one another. For example, if you were to Now consider a much larger population, where there compare the number of differ- are many variants present for ences due to silent mutations each gene, and a relatively between Common and Pygmy large proportion of mutations Chimpanzees, you would find that are varying in prevalence only about one-sixth as many due to genetic drift. After the as you’d find when comparing speciation event occurs, and humans to the Chimpanzee the two new populations family. Since we know that hu- have their own distinct genet- mans diverged from Chimpan- ic makeup, the diversity they zees about six million years began with will cause them ago, this implies that the two to differentiate more easily types of Chimpanzee only and quickly. The gene variants diverged from one another that become common in each around one million years ago. population may have diverged This correlation between si- from one another long before, lent mutation frequencies and and not be the direct result time elapsed since a specia- of the speciation event itself. tion is called the “Molecular Thus, when the Molecular Clock.” Clock is used to determine While we have discussed humans and Chimpanzees the divergence times of a variety of individual genes in the so far, this same technique can be applied to divergences present day, there will be much more variety in the apparent between any two species. Looking farther into our own past, elapsed time since divergence. we can determine that the lineage that eventually led to While the Molecular Clock defines a correlation Homo sapiens diverged from the genus Gorilla approximate- between the total divergence of two species and the time ly 8.6 million years ago, from the genus Pongo (Orangutans) passed since their divergence, a similar correlation exists be- approximately 18 million years ago, and from the family Cer- tween the variance of divergence across multiple portions copithecidae (Baboons, Macaques, Vervet Monkeys) approxi- of a genome and the population size of a common ancestor mately 30 million years ago. at divergence. This allows an estimate to be made of how large the population of a common ancestor was at the time Pi c t u r i n g o u r a n c e s t r a l p o p u l at i o n s of speciation. ust like species and populations have genealogical trees, Jso do individual genes and mutations themselves - they Th e Quantification o f Na t u r a l Se l e c t i o n may splinter, propagate in number or die out just as popu- hile comparing the genes of two related species, ex- lations do. And although two populations come from one Wamining the proportions of silent and non-silent muta- at the time of a speciation event, the mutations and the tions can also yield useful information. As natural selection variants of homologous genes that each population will hasn’t acted on the silent mutations, their number serves as carry with them will come about well before that time. This a baseline, and can give us an idea of how common muta- means that in a manner similar to estimating the amount tions have been in one of our genes over time, regardless of

Spring 2009 13 Evolutionary Genetics

the forces of evolution. The non-silent mutations, however, pare the MYH16 gene in humans to that of other primates, have been affected by selection - what can their propor- we find a moderate number of non-silent mutations - not tions tell us? enough to suggest that the gene is irrelevant, but more than Suppose that a certain gene is vital to human sur- we’d expect to find in the protein of a muscle important to vival, and that any impairment in its functioning will be fa- the survival of the species. We also find that one of them is tal. Natural selection will work very strongly to protect this what we call a “loss of function” mutation, meaning that it gene - because non-silent mutations tend to be harmful, causes the body to produce only a small part of the com- they would be weeded out very effectively, and few would plete protein. In fact, when we analyze the human protein be present. Therefore, we’d expect that the number of fixed itself, we find that some necessary portions of it are com- non-silent mutations would be very small compared to the pletely missing, and that as a whole it’s only about one-third number of silent mutations. the size that it is in other primates. In other words, humans But suppose that another gene has no effect on human have smaller and weaker masticatory muscles, because one health, and is completely irrelevant to our survival - for ex- of the major proteins found in those muscles are small and ample, it has been rendered obsolete by the presence of inoperative. This is not a rare condition - it occurs in all hu- another gene. Non-silent mutations, in this case, won’t be mans. So why would a mutation like this survive, and spread acted upon by natural selection, and they will accumulate throughout the pre-human population? in this gene over time just as silent mutations do - affected When we compare the skulls of humans to those of only by genetic drift, becoming fixed or dying out due simply primates with large temporalis muscles, we find major differ- to chance. Thus, we’d expect that for an irrelevant gene, ences in their shape. As bone develops, it is malleable enough the commonality of silent and non-silent mutations would that its shape may be formed in part by the muscles exert- be roughly equal. ing forces on it, and while primates with very large tempo- Finally, consider a ralis muscles grow, their third gene. Suppose that a As natural selection hasn’t acted on the silent mutations, developing skulls will be population’s environment “their number serves as a baseline, and can give us an idea constrained and stunted has changed such that of how common mutations have been in one of our genes by these muscles. Biolo- this gene not only has no gists hypothesize that benefit, but has become over time, regardless of the forces of evolution. this loss-of-function mu- outright detrimental to ” tation, though it resulted the survival of any individual that carries it. This is a very rare in small and weak masticatory muscles, had the benefit of occurrence, but it is possible that changing circumstances conferring a less constrained and compacted skull, thus al- could cause a gene to suddenly become a liability. What lowing for the expansion of the human braincase and the would we expect to happen? Since non-silent mutations increased brain size that distinguishes Homo sapiens. tend to be harmful to a gene’s function, they would be Under this hypothesis, we would expect that the mastica- actively selected for, and spread throughout a population tory muscles would be preserved by evolution, and non- quickly. So, we’d expect that this gene would have a greater silent mutations would not develop, until the occurrence of proportion of non-silent than silent mutations. a loss-of-function mutation that would be severe enough to Let’s take a real-world example: a gene named MYH16. First, cripple the temporalis muscle and free the brain to devel- put a finger to either temple (on the side of your head, just op in size. After that occurred, non-silent mutations would behind your eye), and clench your teeth as though you were start to appear in the MYH16 gene at the same rate as silent a pre-human primate chewing a particularly tough plant. mutations, as the gene had become irrelevant to survival at The bulge that you feel is the contraction of the tempo- that point. This finding supports the timeline of braincase ralis muscle, which connects your jaw to the side of your development shown by the fossil record - if the gene were head. This is one of the masticatory muscles, which works rendered inoperative in the human lineage approximately 2.4 to close the jaw when you chew your food. You might have million years ago, then a low proportion of non-silent muta- noticed that these muscles are very large in gorillas, which tions in the human lineage before that event and a higher need strength to crush plant matter with their flat teeth, proportion afterwards would result in the middling propor- but are rather small in humans, who have sharper teeth to tion of mutations found in the gene. pierce their food, and require less biting force. While all muscles are made up of long strands of proteins Ne x t St ep s : t h e Ne a n d e r t h a l Ge n o me that pull against one another to create a contracting force, dvances in the field of genetics continue to shed new only the masticatory muscles contain a certain protein, which Alight on the mystery of human development, and the is formed by the MYH16 (myosin heavy chain 16) gene. This pace of progress is continuously growing. In 2006, DNA was protein is particularly large in size, and a muscle that uses it successfully extracted from the femur of a Neanderthal, will be unusually big and strong. However, when we com- estimated to be 38,000 years old, and a new sequencing

14 Columbia Science Review Evolutionary Genetics project began at the Max Planck Institute for Evolution- tion. Proceedings of the National Academy of Sciences. 2002;99:(18) 11736-11741. ary Anthropology in Germany. It’s a challenging task - the Green RE, Krause J, Ptak SE, et al. Analysis of one million base pairs of Neander source material is precious and limited, and the amount of thal DNA. Nature. 2006;444:330-336. contamination by human DNA, which is at least 99.5% iden- Noonan JP, Coop G, Kudaravalli S, et al. Sequencing and Analysis of Neanderthal Genomic DNA. Science. 2006;314:(5802) 1113-1118. tical, is difficult to determine. Patterson N, Richter DJ, Gnerre S, Lander ES, Reich D. Genetic Evidence For Com The project is still underway, and only the first bits plex Speciation of Humans and Chimpanzees. Nature. 2006;441:1103- 1108. and pieces of the genome of our closest relative have been Stedman HH, Kozyak BW, Nelson A, et al. Myosin gene mutation correlates with put online for all to study. In the meantime, however, if anatomical changes in the human lineage. Nature. 2004;428:415-418. you feel the need to do a little research of your own, you Steiper ME, Young NM. Primate Molecular Divergence Rates. Molecular Phyloge netics and Evolution. 2006;41:384-394. can browse completed genomes hosted by the National Takahata, N., 2001 Molecular phylogeny and demographic history of humans, Center for Biotechnology Information (http://www.ncbi. pp. 299–305 in Humanity From African Naissance to Coming Millen nlm.nih.gov/sites/entrez?db=nucleotide) or the University nia—Colloquia in Human Biology and Palaeoanthropology, edited by P. V. Tobias, M. A. Raath, J. of California, Santa Cruz (http://genome.ucsc.edu/). There’s Moggi-Cecchi and G. A. Doyle. Firenze University Press, Firenze, Italy. no telling what you’ll find. Wakeley J, Hey J. Estimating Ancestral Population Parameters. Genet ics. 1997;145:847-855. Wall JD. Estimating Ancestral Population Sizes And Divergence Times. Genetics. References 2003;163:395-404. Wang J. Estimation of Effective Population Sizes From Data on Genetic Markers. Philos Trans R Soc Lond B Biol Sci. 2005. 29;360(1459): 1395–1409 Chou H, Hayakawa T, Diaz S, et al. Inactivation of CMP-N-Acetylneuraminic Acid Hydroxylase Occurred Prior To Brain Expansion During Human Evolu

Expanding the Reaches of the Mind Through Art

Niccola Pérez

rofessor Dalibor Sames has been a faculty member of Columbia’s Department of Chemistry since 1998. PHis research group’s recent research on the neurotransmitter dopamine won the 2008-2009 McKnight Award for Technological Innovations in Neuroscience. Sames and his collaborator, Dr. David Sulzer of the Columbia University Medical School, received the award for their development of Fluorescent False Neu- rotransmitters (FFNs) that allow dopamine to be visualized at individual synapses. Sames elaborates on the temporal and spatial capability of these FFNs, “the brain works by strengthening or weakening synapses, and now we can optically observe activity of hundreds and thousands of individual synapses at the same time.” This technology will enable researchers around the world to further investigate the precise synaptic changes associated with learning and other neurological and psychiatric processes, as well as disorders such as Parkinson’s disease and schizophrenia. But Sames’ exploration of the mind does not stop there. After speaking with a good friend in the art magazine industry about his research on dopamine, Sames realized the possibility for a valuable extension of this research into the art world. More than just a neurotransmitter, Sames sees dopamine as an important element of “fundamental subjective forces like reward, emotion and motivation.” Sames and his friend, Nacho Santos y Gugel, creator of mu, a new Spanish art magazine, have put forth the concept for a groundbreaking art exhibition: an artistic conceptualization of the role of dopamine in animal motivation and action. “It is exciting from a scientific point of view,” says Sames about his work on dopamine, “ but it also opens different perspectives on some philosophical questions, and it is beautiful. I think we have a recipe for an interesting adventure!” The exhibition, currently entitled “A Portrait of the Driv- ing Force of the Animal Universe: The Art and Science of Dopamine,” is tentatively set to debut in Madrid in the spring of 2010. Sames hopes to involve artists from around the world, incorporating a variety of media including photography, video, painting, drawing, sculpture, and text. There are still many decisions to be made, however, each one crucial in shaping the final product. Indeed, one of the largest concerns for Sames is the number of directions he could take the exhibition. Do- pamine does, in fact, play a role in many aspects of animal and human life, from motivation and learning to decision-making and cognition, among others, any of which could be emphasized in the exhibition. Neverthe- less, whichever direction Sames chooses to take the exhibition, it will be a novel and constructive synthesis between the worlds of art and science, one that will certainly shed new insights in both disciplines.

Spring 2009 15 Faculty Profile

{Ch a t t i n g with Ch a l f i e }

Ying Li & Laika Simeon

16 Columbia Science Review Faculty profile

artin Chalfie, Chair of the Department of Bio- people should feel free to take risks. This sort of thinking logical Sciences at Columbia University, shared definitely influenced his later work as he gained the confi- the 2008 Nobel Prize in Chemistry along with dence to take chances and to “think about things in general MOsamu Shimomura and Roger Y. Tsien “for the terms and not be afraid to make strange connections.” discovery and development of the green fluorescent pro- Unsure of what to do after graduation, a sentiment tein, GFP.” He has received a great deal of much-deserved many students can identify with, Chalfie worked at a series attention lately and was one of the 49 American Nobel lau- of short-term jobs and eventually ended up teaching chem- reates who wrote to President Obama, urging for an increase istry in a high school outside New Haven. One advantage in funding for scientific research. We were lucky enough to of the job was that his summers were free to pursue other sit down and talk with him about his life and his education interests. Before the start of one summer a colleague of in the sciences. He was funny, warm and incredibly down- his recommended that he work for Jose Zadunaisky, who to-earth despite the fact that he had just finished another headed a lab at Yale University. During his meeting with Za- interview and his phone was ringing off the hook. We were dunaisky, Chalfie learnt that the lab was studying chloride all enraptured by his candid account of his time as an un- transport in the cornea of frogs by using the Ussing chamber. dergraduate and his evident passion for scientific research. The apparent similarities between the lab he was interview- While gesticulating excitedly, he told us about how he came ing to work in and the research he had written about for his to be a scientist as well as entertaining anecdotes about his undergraduate cell biology class prompted him to ask Za- experiences along the way. dunaisky whether cyclic AMP was involved. Zadunaisky did Chalfie started out at not know the answer to his Harvard in 1965, planning to question as no experiments major in mathematics but af- Be i n g a n undergraduate had been conducted before. ter taking an advanced calcu- “ t a u g h t h i m t h a t t h e r e r e a l l y Chalfie was hired and assigned lus class, discovered he “didn’t to do “the world’s most bor- quite have the aptitude for it.” i s n o o n e r i g h t a n s w e r a n d t h a t ing project to do for a summer He then turned to biochemis- project”. But his boredom was try which was at that time a p e o p l e s h o u l d f e e l f r e e t o t a k e quickly dissipated when Za- “sassy topic” with many new r i s k s . dunaisky left for the summer and exciting developments. It and Chalfie was on his own. also allowed his math classes ” He talked to the postdocs to count towards his major as well as take more science and graduate students about the possibility of investigating classes including physical chemistry and biology. In his ju- whether cyclic AMP was involved in the specific pathway nior year he took a cell biology course that was essential the lab was studying. With their help, he managed to set to his decision to go into science. He wrote an essay for the up an experiment in which he proved cyclic AMP did indeed class about experiments involving sodium transport in toad increased chloride transport. When Zadunaisky returned to bladders that used the Ussing chamber to measure short the lab at the end of the summer and asked to see the circuit current – the current needed to counter the amount data from the experiment he had assigned to Chalfie, he of ion transport. And most importantly, it was a mechanism was surprised to see that Chalfie had succeeded in doing an that was activated by the small second messenger molecule independent experiment. The result of his work that sum- cAMP (cyclic adenosine monophosphate). Chalfie enjoyed mer was his first scientific publication and the motivation to the class so much that he decided to work in a lab the sum- go on to graduate school to pursue research. mer after his junior year. But he was a “complete and utter The main difference between what happened in the failure” as he repeated one experiment 10 times and failed lab after his junior year and what happened two years later miserably every time. The experiment required a great deal was that he had the help of the other people in the lab. He of preparation and the experience was very discouraging. By didn’t feel as though he had to do everything on his own. the end of the summer, Chalfie had decided that science During his first lab experience, he would have felt less dis- was not for him and so during his senior year he finished up couraged and more inclined to continue in the sciences had his major requirements and took a wide variety of courses in someone told him that most experiments never work out Russian literature, theatre and law. and that he shouldn’t worry. Instead he felt as though he When asked what the most important thing he was failing. However in the Zadunaisky lab he found many learnt from these undergraduate years, Chalfie replied, “the people he could talk to and succeeded in doing an experi- importance of having a bunch of facts and being free to put ment. He also credits his undergraduate training for giving those together in different ways”. Being an undergraduate him the nerve to ignore what he was assigned to do and do taught him that there really is no one right answer and that his own experiment.

Photo by Ivy Chen Spring 2009 17 Faculty Profile

Chalfie continued his studies at Harvard University calcium ions and GFP absorbs this light and then emits green and enrolled in the physiology department in 1972. He cred- light. In the 1980s at the Woods Hole Oceanographic Insti- its his mentor, Robert Perlman, for being a supportive and tution, Douglas Prasher first realized that GFP could be used encouraging figure with whom he could talk “about any as a tracer and set out to find the genetic sequence for the idea no matter how silly or stupid it was.” During graduate protein. Columbia University professor, Martin Chalfie con- school, Chalfie studied the synthesis and release of adrena- tinued Prasher’s work and was able to incorporate GFP into line. When he was looking for postdoctoral positions his E. coli bacteria and C. elegans, roundworms. high school friend Bob Horowitz, who was a postdoc in Syd- GFP has ney Brenner’s lab at the Laboratory for Molecular Biology in since been used in Cambridge, United Kingdom, encouraged him to apply for a many other organ- research position in Brenner’s lab. Chalfie liked the research isms. It is very small, that was going on in Brenner’s lab and thought that going off doesn’t require co- to another country for a few years was a wonderful idea, so factors, and doesn’t he applied and started working there in 1977. Chalfie praises harm the struc- Brenner as being “a dazzling marvel of a scientist” and he tures in which it is believes that his time as a postdoc was “the most formative expressed. GFP can part of [his] scientific life.” He recalls that being a postdoc also be expressed in was “really a wonderful time, you’re not taking any courses, many different col- you’re not taking any exams, you’re just doing research. You ors besides green. don’t have any obligations that professors have…your time Cyan, red, yellow, is your own.” And the Laboratory of Molecular Biology pro- and orange fluores- vided Chalfie with the perfect environment to do research cent proteins have been developed. More recently Brainbow, as he was given absolute freedom and all the facilities and a technique that incorporates many colors to tag hundreds supplies for experiments. of neurons at the same time, has been developed at Harvard When asked what makes a good scientist, Chalfie University (the cover of this issue is of these brainbow neu- hesitated before saying that originality, a willingness to do rons). GFP has also been used for detecting the activity of something different and dedication were important. But a gene promoter. This is possible because GFP and the gene when asked how one becomes an amazing scientist, he mod- in question are expressed together by the same promoter. estly said he has no idea. Whatever the gene codes for in the cell will then fluoresce. Photoactivatable fluorescent proteins have also been devel- oped. These proteins are able to be hyper fluoresce, which A Background on GFP enable researchers to use high energy beams to track in- dividual cell, organelle, or protein movement. Fluorescent proteins also enhance the ability of FRET (Fluorescence Resonance Energy Transfer), a technique to track interaction between proteins. Regardless of the specific technique, GFP has already revolutionized our understanding of molecular and cellular biology, and become a fundamental tool used in many fields of science. It continues to play a crucial role in a great deal of research as it is used more and more cre- atively, as well as more extensively.

References

Chalfie, Martin, Yuan Tu, Ghia Euskirchen, William W. Ward, and Douglas C. Prasher. Science 263. Chudakov, Dmitriy M., Sergey Lukyanov, and Konstantin A. Lukyanov. “Fluorescent proteins as a toolkit for in vivo imaging.” Trends in Biotechnology 23 (2005). Prasher, Douglas, Virginia K. Eckenrode, William W. Ward, Frank G. Prendergast, and Milton J. Cormier. “Primary structure of the Aequorea victoria green-Williamscent protein.” Green Fluorescent Protein (GFP) is one of the most widely Gene 111 (1992). used proteins used in research today. In 1978, Prendergast Prendergast, Franklin G., and Kenneth G. Mann. “Chemical and Physical Properties of Aequorin and the Green Fluores and Mann at the University of Minnesota isolated GFP from cent Protein Isolated from Aequorea Forskilea.” Biochem a luminescent jellyfish, Aequorea Victoria. They realized that istry 17 (1978). another protein, aequorin, emits blue light when exposed to

18 Columbia Science Review Picture courtesy of the Tsien Lab Sleep

the [Wonderful World of Sleep] Anna Corke magine you’re exhausted after a long night of studying. can also complicate things. Scientists know that metabolic IYou stumble to your 9 a.m lecture, eyes hardly open. You activity in the brain is greatly reduced during one type of try to stay focused as the professor begins to speak. You get sleep, called slow-wave sleep. But another type of sleep, the feeling that maybe you’re not quite there. But, you’re Rapid Eye Movement sleep (REM) is associated with an convinced you haven’t fallen asleep. Or have you? increase in metabolic activity in the brain. Most brain regions Scientists are realizing that one of best way to look like they’re awake in REM. In humans, at least, we find out what sleep does is to research the consequences know enough about behavior to be able to tell REM sleep of its absence. Some scientists study the brains and from the awake state. But when tapping into the brains of behaviors of sleep-deprived humans. Others are looking at animals, it’s difficult to correlate brain activity with specific sleep deprivation in animals and are encountering startling behaviors since telling whether animals are asleep or not—or disparities in the nature and function of sleep between whether they need to sleep in the first place—is much more humans and animals. difficult. As college In order to explore students all find out, sleep whether and how animals is vital to maintain health sleep, scientists are and sanity—in humans examining what happens at least. In addition when animals are sleep to the physical and deprived. Since scientists psychological problems can’t tell just by looking caused by loss of sleep, whether animals need to Dr. Yaakov Stern, Professor sleep, they keep them of Clinical Neuropsychology at Columbia University’s Taub awake and study the consequences. One study, in which rats Institute, said that “people who are sleep deprived can were prevented from sleeping for several weeks, found that have brief bouts of sleep that they are unaware of.” With the rats died after just a few weeks of sleep loss. However, enough sleep-deprivation, we can become unaware of our Dr. Siegel reports, some other studies have shown less unconsciousness. Dr. Stern pointed out that this can cause extreme consequences of sleep deprivation. For instance, major problems, for instance, “in a situation that requires keeping pigeons from sleeping doesn’t seem to affect continuous attention, such as driving.” them at all. Studies in cockroaches have found that these Scientists who study sleep in animals are finding species have resting states but do not seem to accumulate it more and more difficult to define. Merriam Webster’s “sleep debt”—the need to sleep more to make up for lost definition, “the natural periodic suspension of consciousness sleep—in the way a human might after an all-nighter. during which the powers of the body are restored,” is simple One species of frog, Hyla septentrionalis, sleeps, compared to what the definition would be if it took into while another, Rana catesbeiana, doesn’t. At account animals’ various ways of “sleeping.” Now most the same time, birds have REM and slow- scientists believe that sleep must be defined at the species- wave sleep states just like humans. specific level. Another way scientists For instance, imagine a dog lying on a porch in the approach the sleep problem sun. His eyes are closed and he is breathing deeply. Is he is to look at animals that asleep? In the recent review, “Do all animals sleep?” that seem to be chronically appeared in Trends in Neurosciences earlier this year, Jerome sleep-deprived by M. Siegel, Director of the Center for Sleep Research at their normal UCLA, discusses the problems scientists are having with behavior. For answering that question. For instance, humans can sit quietly instance, the with their eyes closed and not be in the state of reduced Swainson’s sensory awareness that is associated with sleep. Just looking t h r u s h , and acting asleep doesn’t mean an animal is. C a t h a r u s Similarly, looking at brain activity using ustulatus, travels electroencephalographic recordings, or EEG, during sleep around 5000 km from

Picture courtesy of the Tsien Lab Spring 2009 19 Sleep

Canada to Mexico or South America and back each year—an What would it be like to sleep with just one exhausting ordeal that leaves little time for sleep. During hemisphere of the brain? Dr. Siegel suggested in an email seasons when they aren’t migrating, these birds sleep at that “if humans had slow waves in one half of the brain they night. But a recent study found that during the migration would presumably be quite impaired, even half-paralyzed.” months, Swainson’s thrushes fly at night and take short naps But he went on to say that “dolphin brains have evolved to during the day to make up for the loss of sleep. function at a high level during this behavior, so it is unclear But these are not your average naps. Some birds, if there is any substantial deficit.” including the Swainson’s thrush, are able to sleep in just one Whether humans could sleep in this way—say, during half of the brain at a time. The phrase, “sleep with one eye that 9AM class— is probably “more science fiction [than open,” takes on a whole new meaning for birds: keeping an reality],” Dr. Stern pointed out. But discovering that many eye on one’s surroundings while taking naps during the day animals do not need the kind of sleep that humans do has can mean the difference between life and death. been changing how scientists think about sleep’s purpose in Called “unihemispheric sleep” by scientists, sleeping the brain. Knowing that REM sleep has not been observed in in one hemisphere of the brain has also been observed in dolphins has led some to believe that unihemispheric sleep aquatic mammals, such as the fur seal, Callorhinus ursinus. occurs in lieu of REM. “This suggests that REM sleep is not A study found that when fur seals are on land they sleep necessary if half of the brain is always active,” Dr. Siegel just like terrestrial mammals. However, when fur seals are said. “REM may serve to stimulate the brain in sleep.” Exactly in the water, their sleep patterns change: their brains exhibit why our brains need this stimulation, while other animals’ regular slow-wave sleep in one hemisphere and waking state brains do not, is still a mystery. Perhaps further studies of EEG patterns in the other. Paddling with one fore flipper and sleep deprivation in animals will help scientists uncover the floating on one side, they keep one eye open, watching their reasons behind our own sleep patterns. surroundings and where they are headed, while they sleep in References the other half of the body. Other aquatic mammals, such as dolphins, only sleep Merriam Webster Collegiate Dictionary, 10th ed. with one half of the brain at a time, too. However, they Fuchs T, Haney A, Jechura J, Moore FR, Bingman VP. Daytime naps in night- migrating birds: behavioural adaptation to seasonal sleep deprivation don’t exhibit any of the asymmetrical body movements in the Swainson’s thrush, Catharus ustulatus. Animal Behaviour. 2006. of the fur seal. Dolphins seem to be able to swim while 72:951-958. Lyamin, OI, Lapierre JL, Kosenko PO, Mukhametov LM, Siegel JM. sleeping in one half of the brain just as easily as humans Electroencephalogram asymmetry and spectral power during sleep in breathe while sleeping. the northern fur seal. J. Sleep Res. 2008. 17:154-165. Siegel, JM. Do all animals sleep? Trends in Neurosciences. 2008. 31(4):208-213.

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20 Columbia Science Review Perspectives Blue Brain Blue Brain

Elizabeth L. Robinson In 1997, IBM’s Deep Blue supercomputer defeated world plexity arises more from its wiring than from its actual com- chess champion Garry Kasparov. For many, this monumental ponents. Neurons, though they generally share the same match up between man and machine was proof that, at least basic properties, number in the billions. The problem, when when it comes to calculations, computers had us beat. The it comes to simulation, is that each of these neurons com- likes of Hal from 2001: A Space Odyssey seemed almost municates with hundreds or even thousands of others and inevitable. However, a group of scientists in Switzerland are as of now, there is no computer powerful enough to pro- playing a new game with Deep Blue’s progeny, another IBM cess all of this information at once. At half speed, it takes supercomputer called Blue Gene, and this time humanity has a computer the size of several refrigerators to model only the upper hand. .00001% of a human brain. Nevertheless, since the problem The project is called Blue Brain. The goal is to build is one of volume rather than complexity, it seems likely that a functional model of the human brain from the ground up. technology will eventually catch up. This process of modeling, though common in physics, is an In as little as two years, the Blue Brain Project ex- entirely new approach in the field of neuroscience, which pects to be able to create an entire rat brain. They are al- generally studies the brain by taking it apart and observing ready working with a robotics company in Japan to design the basic biological processes. The Blue Brain project is using a robotic rat that can embody their model brain. The big this same biological data to actually put one together. By question is: what will happen? Will the rat be “alive” and merging biology, computer science, and design, they hope to “conscious?” open a new window into the brain. The stated goal of the project is to create an ac- If Blue Brain becomes a reality, scientists will be able to curate model of the brain for testing purposes in neurologi- study the brain with unprecedented accuracy and control, cal studies. However, the implications of actually building a repeating and perfecting experiments before performing brain are vast. If the project is successful, it could answer them in the lab. This technology would also lead to a bet- one of the greatest questions of science and philosophy: ter understanding of the brain by allowing scientists to simu- what does it mean to be human? If intelligence is entirely late brain processes, disorders, and injuries and observe their biological, a biologically accurate model of the brain should properties on the neuronal level. be conscious. Henry Markram, director of the Blue Brain proj- The project uses IBM supercomputers and powerful ect says, “I think it will be just as interesting, perhaps even simulation software to create detailed 3-D representations more interesting, if we can’t create a conscious computer. of neurons that function and communicate just like the real Then the question will be: ‘What are we missing? Why is thing. The computer itself is actually designed like a brain this not enough?’”(Lehrer). Still, it will take many years and in that it consists of thousands of nearly identical, discrete huge technological advances to make this project a reality. units, like neurons. Also, the components of this computer At least for now, the human brain is still the most powerful strengthen their connections the more they communicate. computer around. (In a regular computer, the strength and speed of connec- tions is static across the entire system.) The data from Blue References Gene is then collected and processed by imaging software that can form navigable pictures of the simulated neurons. Markram, Henry. “The Blue Brain Project.” Nature Reviews Neuroscience 7, 153- 160. February 2006. In this way, inscrutable data becomes readily comprehen- http://bluebrain.epfl.ch/ sible. Lehrer, Jonah. “Out of the Blue.” Seed Magazine. http://seedmagazine.com/ The brain is extremely complex. However, this com- news/2008/03/out_of_the_blue.php

21 Picture courtesy of Nikon Small World Spring 2009 21 Tower of Hanoi Tower

1. History of the Puzzle The Tower of Hanoi game is a historically famous mathematical puzzle, which was first invented by the French mathematician Édouard Lucas in 1883. The game con- sists of three wooden pegs and a certain number of disks, each different in its size and all stacked on a singlepeg ini- tially in order of size with the biggest at the bottom and smallest at the top. The problem is simple, yet interesting of size with the biggest at the bottom and smallest at the top. The problem is simple, yet interesting: what is the mini- mal number of moves needed in order to move all disks So-Eun Park to another peg and stack them again in order of size? The players, however, must go by two rules: first, only one disk on top of a peg is moved at a time; second, no disk can be stacked on a smaller one. The puzzle relates its origin number of bikes each man bought. Of to a religious legend about a Vietnamese or Indian temple course, those who are mathematically which contains three posts and 64 golden disks on them. savvy can come up with a simple solu- The monks started moving the golden disks at some point tion to this sequence problem to save in the past, one per second under the rules specified above, the tedious computations. and the legend says the world will end when the moving is The Fibonacci sequence is completed. Now, what would you estimate the lifetime of one example of a famous recursive the world? With only 64 disks, does it seem like it would algorithm. The Italian mathematician end sometime soon? Well, the truth is that you don’t have Leonardo of Pisa (known as Fibonacci) to worry about this old myth at all, for roughly the next introduced this sequence in his book written in 1202. The 555 billion years! first two terms of the Fibonacci sequence are set to be 1, and starting with the third term, each consecutive term is 2. Recursive Algorithm defined to be the sum of the previous two terms right be- Here comes the logic underlying this puzzle that fore it. Therefore, the sequence is1,1,2,3,5,8,13,21,34,55,89,1 mathematicians call a “recursive algorithm.” Recursive algo- 44, and so on and so forth. We can formulate this into a rithm is a mathematical method used to find solutions to a mathematical statement: question by using already given or found solutions to previ- ous questions. An easy example is to calculate the num- a _(n)=a _(n-1)+a _(n-2), ber of bikes one man buys in relation to the number of where a _(i) indicates the ith term and a _ 1 = a _ 0 = 1. This bikes previous men buy before him. If each man buys twice implies that we can find the nth term once we know what as many bikes plus one more than the previous man then the n-1st and n-2nd terms are, each of which we can find by we could calcuate the number of bikes one man will buy, calculating the n-3rd and n-4th terms, and recursively the given an initial purchase. Therefore we can calculate how n-5th and n-6th terms, and so on. In short, any particular many bikes the fifth man will buy, when the first man buys term can be deduced once we know all previous terms. As only one bike. In order for us to calculate the number of in both examples, the characteristic of these mathematical bikes the fifth man will buy, we must know the number of solutions that involve previous steps in finding the answer bikes the fourth man will buy (so that we can double it and for the next step is called recursive. add one to get the answer); but to get the number for the fourth man, we must again know the number of bikes the 3. Solution to the Tower of Hanoi Game for Everyone third man bought. Recursively, we would have to know the Amazingly enough, the solution to the Tower of 22 Columbia Science Review Tower of Hanoi

Hanoi problem is easy to find with the help of a recursive requires a _ 2 number of moves by the symmetry of pegs. algorithm and a little bit of modifications. Here, we will de- If these final moves are done, all three disks will be all fine each nth term, a _ n, of the sequence (that is, the nth stacked on peg 3 as we wished. This piecewise understand- solution to the Tower of Hanoi game as we defined the nth ing of the problem will give us a recursive equation: man and the nth Fibonacci term in section 2) as the solution to the game with exactly n disks. We also want to label a _ 3 = 2* a _ 2 + 1 each peg: 1, 2, and 3, in order to make this section more which can be extended to general cases for any number of readable notation-wise. Then, assume that in all cases, all disks: disks are initially stacked on the peg 1 in size order and we a _ n= 2* a _ n - 1 + 1 want to move them all to peg 3. By a simple mathematical computation, we get the answer: Therefore, a _ 1 will be the minimal number of a _ n = 2^ n - 1 moves needed to move a single disk from peg 1 to peg 3, Therefore, the answer to the original question and the answer is obviously one! Next, think about a _ 2, about the world’s lifetime is 2^64-1 seconds, which is ap- the case for 2 disks, which also easy to figure out. First, we proximately 600 billion years. will move the smallest disk on top to peg 2, then the larg- est disk to peg 3 that just became free by the smallest disk 4. Unanswered Variations moving away. By placing the smallest back on the largest In fact, the Tower of Hanoi problem with 3 pegs is disk on peg 3, we can stack all two disks on peg 3 in order an easy problem to solve for all of those who are interest- of size as desired by ed in mathematics only 3 moves. or mathematical From a _ 3, puzzles . the recursive algo- Then why rithm will finally begin is the Tower of to be used and lead Hanoi game called us to find the an- the “long unsolved swer in the simplest mystery”? Amaz- way possible. For us ingly enough, once to move all disks to the number of peg 3, we need to pegs goes over at some point in the three, the solution middle move the to the problem largest disk to peg 3. becomes unbeliev- This indicates that at ably complicated the moment the other and the only discov- 2 disks will be stacked on peg 2. This is because, by the rules ered “algorithm” (not a solution) that is most likely to be of the game, only the top disk on each peg can be moved the answer, which is called the Frame-Stewart Algorithm, is and any disk cannot sit on a smaller one. On top of that, the closest we have gotten so far to the general solution. the other 2 disks will have to be stacked on peg 2 in order Moreover, there are interesting variations to this of size. question as more restrictions on the moves can be added Now, let’s look at this procedure backward from (ex: labeling disks with numbers and not allowing odd num- this point on. The largest disk is on peg 1, peg 3 is empty to bered disks to sit on odd numbered disks; only allowing receive a visit from the largest disk, and the other 2 disks disks to be moved to the right) to looking at this problem are stacked on peg 2. How can this stage be reached from from more mathematical angles (ex: defining a graph of the the initial state in a minimal number of moves? Yes, only by game, applying graph properties to the game) to extending repeating exactly what we did to get a _ 2. The 2 smallest the minimal number of moves (shortest path) concept to disks were initially on peg 1 on top of the largest disk, but between any states of disks on pegs (ex: we don’t have now they moved their peg and are sitting on peg 2 (Moving to begin with all disks on a single peg, what is the minimal them to peg 3, as in the a _ 2 procedure, or to peg 2 require number of moves needed to stack all back to a single peg the same number of minimal moves because they are sym- if they were initially all spread out?). metric). Hence, up to this point, the moves require a _ 2 We can all name many different kinds of restric- number of moves. Then, we move the largest disk to peg tions off the top of our heads and make our own Tower of 3, which adds one more move (it is a _ 2 + 1 now). What we Hanoi variation problem. If it is challenging enough, you will have to in the next step is to move the smallest 2 disks on have an open problem known to the world, like Fermat’s peg 2 back to the top of the largest disk on peg 3. This again last theorem, named after you!

Spring 2009 23 G6PDD The Curious Case of the Glucose-6-phosphateBeans dehydrogenase deficiency :

Amy Huang 44 year old Ashkenazi Jewish woman with “a five day identified other factors such as the intake of various antima- Ahistory of feeling tired and unwell” and “bouts of vom- larial agents, sulfonamides, acetaminophen, anthelmintics, iting” is rushed to the emergency room. She denies any drug non-steroidal anti-inflammatory drugs (NSAIDs), naphtha- use, foreign travel, or previous cases similar to the one she lene, quinine, Thiazide diuretics, nitrofurantoin, and the fava is presenting. She displays hypertension and jaundice, a con- beans that give the disorder its characteristic name. dition in which bilirubin builds up to produce a yellowing of Glucose-6-phosphate dehydrogenase, a polypeptide the skin, sclera and the whites of the eye, and dark urine. of 515 amino acids, is crucial for the production of NADH, She denies a family history of anemia and neonatal jaundice. a cellular coenzyme, in the hexose monophosphate shunt, However, she states that she had eaten a whole pound of an important metabolic pathway. D-glucose-6-phosphate broad beans shortly before her symptoms started. A case must be dehydrogenized by D-glucose-6-phosphate dehy- of the food allergies? Not quite. Her blood smear shows ex- drogenase into 6-phosphogluconolacton and NADH. The tensive red blood cell damage and irregularly-shaped cells, NADH produced maintains the level of glutathione in the atypical results for a mere allergy. What was in those beans cells, which protects erythrocytes from oxidative damage. that made her display these severe symptoms within such a A deficiency in glucose-6-phosphate dehydrogenase severe- short period of time? ly limits the level of glutathione and NADH, creating a high risk for hemolytic anemia. The consumption of foods rich in Beans, aspirin, and liver problems? oxidants such as the vicine, divicine, convicine and isouramil Broad beans contain a toxic chemical called vicine, which chemicals in broad beans, can bring about oxidative stress oxidizes red blood cells, causing hemolytic anemia, a condi- and hemolysis in patients. tion in which red blood cells burst. Nevertheless, most peo- Other biological elements like proteins are damaged ple who consume broad beans do not display such symp- in oxidative stress, as all glutathione are consumed during toms, except for those who lack the enzyme to protect the pathway. The oxidation of proteins causes electrolyte their erythrocytes from oxidative damage. This enzyme is imbalance and hemolysis, leading to acute renal failure, glucose-6-phosphate dehydrogenase, and those who have hemoglobinuria, hemolytic anemia, and jaundice from the a deficiency of it have glucose-6-phosphate dehydrogenase metabolism of hemoglobin into bilirubin. Clinical manifes- deficiency (G6PDD). Though it seems severe, this condition is tations include splenomegaly, hepatomegaly, confusion, surprisingly common. G6PDD, or favism, is one of the most fever, fatigue, pallor, dark urine from the filtering of lysed common enzyme deficiency disorders in the world. It af- erythrocytes, tachycardia, dyspnea, and palpitations. Some fects more than 400 million people, with higher prevalence symptoms of G6PDD are present after birth, with persistent in Africa, Asia, Southern Europe, and Oceania. Although it is a neonatal jaundice being the most prevalent. The low levels commonly inherited disorder that dates back to 5th century and inefficient activity of G6PD in the liver result in the yel- BC, G6PDD was not widely studied until the early 1950’s, low discoloration of the mucous membranes, skin, and the when the US Army conducted research on the etiology of whites of the eyes. If left untreated, there can be death the high rates of drug-related sensitivity in black soldiers. It or permanent neurological damage. In G6PDD individuals, was found that the antimalarial drug, primaquine, was re- a hemolytic crisis ensues from the intake of certain medi- sponsible for up to 15% of the cases, while later studies cations and foods rich in oxidants. The buildup of oxygen 24 Columbia Science Review G6PDD chemicals in erythrocytes destroys the cells by lysis, caus- G6PDD patients can lead to hemolysis in two cases out of ing anemia. forty. Although there is controversy over the validity of the data, G6PDD patients should limit or avoid the intake Diagnostic Techniques of aspirin. Many diagnostic tests are available for the testing of G6P- DD, such as the methylene blue test and the methemo- Epidemiological Significance globin reduction test. In the methylene blue test, a cath- G6DD is of epidemiological significance, in that it offers eter is placed into a vein and methylene blue, a dark green some resistance to Blackwater fever, or malaria caused by powder, is placed into the catheter. The health care provider Plasmodium falciparum. This relationship is similar to that of will observe how the powder will turn methemoglobin into sickle cell anemia, in which erythrocytes take on a distinct hemoglobin in order to determine the type of methemo- crescent form. The mechanism of action for this resistance globinemia, in which several types of such have genetic is that the spleen rapidly clears out the malaria parasite for causes. The methemoglobin reduction test is a blood test G6PDD patients, offering them an evolutionary advantage determines the ratio of normal hemoglobin versus hemo- against a virulent disease. Since the erythrocytes are abnor- globin derivatives (methemoglobin, carboxyhemoglobin, and mal, the malaria parasite has a difficult time attacking them. sulfhemoglobin). Abnormal values of hemoglobin derivative Researchers have also found that G6PDD may levels may indicate inefficient transport of oxygen to cells, be linked to cataractogenesis, Fulminant Rocky Mountain as the derivatives are unable to carry oxygen. Another test, spotted fever (RMSF), cancer, and possibly cystic fibrosis the Beutler fluorescent spot test, looks at G6PD enzyme and diabetes. A study among Saudi-Arabian patients found activity under UV light at 30°C. Elevated bilirubin and serum that G6PD activity in diabetics is much lower than in non- LDH levels, low serum haptoglobin and hemoglobin count, diabetics. Another study illustrated how insulin deficiency hemoglobinuria, elevated absolute reticulocyte count, and was proportional to decreased G6PD activity. the presence of Heinz bodies in a blood smear can also in- dicate G6PDD. The Genetics of G6PDD G6PDD is an X-linked recessive disorder resulting from a Treatments and Prevention single-point mutation of a substitution of an amino acid Although there is no known cure for G6PDD, various treat- coding for the G6PD structural gene at the Xq28 region ments alleviate symptoms of the patient. For infants with located on the tip of the long arm of the X- chromosome. neonatal jaundice, special lamps called bili-lights help It is inherited from heterozygote females who carry the relieve the jaundice by breaking down the bilirubin accu- gene on one X chromosome, who then passes it to her off- mulated in the skin and mucous membranes (since bilirubin spring, in which the sons who carry one copy of the gene is readily decomposed by light). However, this treatment is affected, and the daughter who also carries one copy is does not cure the underlying condition of jaundice. a carrier. Thus, males are more likely to be affected than In the case of acute hemolytic anemia, a blood females. The severity of the disorder is affected by race, transfusion may be necessary. However, the simplest as Africans who carry the disorder exhibit a milder form of procedure is to treat the patient with nasal oxygen and hemolysis than do Caucasians. People with G6PDD should adequate bed rest to alleviate symptoms. Folic acid supple- consult genetic counselors to assess their disorder and to ments and treatment with human haptoglobin products can determine the risk of passing it on to their children. also be administered. In severe cases, patients can undergo a splenectomy to lessen the symptoms of anemia, as the Potential Cures spleen is the main organ in erythrocyte destruction. None- Future treatments of G6PDD are limited, but can involve theless, the best way to prevent a hemolytic crisis is to genetic engineering of certain food items containing high avoid eating broad beans and taking certain medications. Pa- amounts of oxidative chemicals in order to prevent the tients should consult their physicians before they attempt onset of hemolytic anemia in people who consume these to take medications and supplements. It is noted that staple foods, especially for people in Middle Eastern and some patients can tolerate small amounts of certain drugs, African countries. Scientists are attempting to produce ge- depending on the severity of the disorder. For patients with netically engineered fava beans that lack oxidative agents, acute renal failure, dialysis offers a temporary solution to a a move that can benefit Saudi Arabians, whose rates of severe problem. favism are high even though fava beans remain a food Infections can also precipitate into a hemolytic staple. crisis, and patients are advised that many medications that A potential treatment proposal may be the intro- are normally safe to administer may lead to hemolysis. In duction of potent antioxidants into the bodies of G6PDD a study done in Algeria, it has been discovered that even patients. Although it is difficult to always maintain an ade- the ingestion of low doses (400mg) of acetaminophen in quate level of glutathione in the cells, it might be possible

Spring 2009 25 G6PDD

to put these antioxidants in slow-release capsules to help accounting for the jaundice and hepatomegaly.

prevent oxidative stress and hemolysis when consuming References foods. One method is to encapsulate them in kelp deriva- Lim, F., Vulliamy, T., and Abdalla, S.H. (2005).An Ashkenazi Jewish woman present tives for slow release, a technique some researchers found ing with favism. Journal of Clinical Pathology. 58(3): 317–319. useful for the delivery of certain drugs. One solution can Beutler, E. (1991) Glucose-6-phosphate dehydrogenase deficiency. New England be to use potent antioxidants to reduce drug dosage and Journal of Medicine. 324:169–174. Matsui, W (2007). Glucose-6-phosphate dehydrogenase deficiency. Retrieved Oc lessen intake frequency. Research from the Salk Institute tober 28, 2008, from MedlinePlus Medical Encyclopedia Web site: of Biological Studies and Columbia University shows that http://www.nlm.nih.gov/medlineplus/ency/article/000528. nanoparticles, particles composed of yttrium and cerium htm#Alternative%20Names Kunst, A. et al. (1984) Methods in Enzymatic Analysis (Bergmeyer, H.U., ed.) oxides, can potentially become the most potent antioxi- 6(3):163-72 dant used to counteract severe oxidative stress. However, Beutler, E. (1991) Glucose-6-phosphate dehydrogenase deficiency. New England Journal of Medicine. 324:169–174. the cytotoxicity of such particles is undetermined and the Kirmse, B (2006). Methylene blue test. Retrieved October 28, 2008, from Med chemicals are possibly dangerous to humans. If discovered linePlus Medical Encyclopedia Web site: http://www.nlm.nih.gov/ to be safe for use, they can help improve the lives of medlineplus/ency/article/003412.htm Markic J, Krzelj V, Markotic A, et al (August 2006). “High incidence of glucose-6- thousands of G6PDD patients. Although these treatments phosphate dehydrogenase deficiency in Croatian island isolate: ex may not prove feasible, they are only ideas that can per- ample from Vis island, Croatia”. Croatian Medical Journal. 47(4):556– petuate the study for the finding of better treatments for 70. Benabadji, M. et al. (1978).Heterogeneity of glucose-6-phosphate dehydrogenase G6PDD. deficiency in Algeria. Human Genetics. 40(2):177-184. One potential cure may be to find a chemical Sobngwi, E. et al. (2005). High Prevalence of Glucose-6-Phosphate Dehydroge nase Deficiency without Gene Mutation Suggests a Novel Genetic alternative to glutathione and infuse it into the body of Mechanism Predisposing to Ketosis-Prone Diabetes. The Journal of the G6PDD patient, or to find a chemical to help maintain Clinical Endocrinology & Metabolism. 90(8):4446-4450. glutathione levels. Pharmaceutical research helps advance (2004). G6PD. Retrieved October 28, 2008, from University of Virginia Health system http://www.healthsystem.virginia.edu/uvahealth/adult_blood/ the search for such a drug that can replace NADH as the glucose.cfm chemical that balances glutathione levels. These drugs (2006) Nanotechnology to design potent antioxidants. Retrieved October 29, should target the liver and enhance the activity of G6PD 2008, from Nanowerk website: http://www.nanowerk.com/spotlight/ spotid=1114.php there, as G6PD activity is lowest in that organ in patients,

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26 Columbia Science Review Delian Problem The Delian Problem

Atanas Atanasov We are allowed to use an unruled edge and a com- pass. The term “edge” is adopted instead of “ruler”, in order to indicate an idealized nature, namely that it has only one side and extends infinitely. What is often omitted from the 1. Introduction description is the availability of a marking device, say a pen- cil, using which we can draw lines along the edge (we use During a rigorous geometry course in middle school draw as a synonym for construct). Similarly, the compass or high school one often encounters various construction is a device such that given two points, we can construct a problems. We are customarily asked to construct or extend circle having the former as a center and the latter lying on a chart using straight edge and compass alone. For example, the circumference of the circle. It is important to note, once it is useful to build the median or the altitude of a triangle lifted the compass collapses. In other words, we cannot use and other such. Most of these problems bear their roots it to transport lengths as one would often use such a device in the ancient world. By no means we will attempt a com- in real life. If this were allowed, then trisecting the angle plete historic assessment of such constructive geometric is possible. It is by no means apparent why these are the problems, but only give the flavor of the general picture. set of construction axioms we abide by. This set of rules turns out to correspond to a tractable algebraic model. All problems previously mentioned may be resolved by elementary means – most people should have no trouble If we are asked to construct a line segment with constructing a solution in a very short timespan. Mathemati- a prescribed length, only given a straight edge and com- cians, on the other hand, take equal pleasure in asking ques- pass, there is no indication of scale. That is, a segment is tions of existence and inexistence. It is natural to try iden- of length two, if and only if it is twice a segment of tifying ob jects which we cannot construct. A well-known length one. For this purpose of scaling, we assume there legend tells the story of a plague in Athens during the 5th exists a coordinate system in the plane all constructions BCE. After consulting the oracle of Apollo at Delos, a deci- take place. Simpler put, there is a marking on the edge sion has been taken to double the size of the altar devoted that indicates the unit length (this enables us to con- to Athens, which was roughly the shaped of a cube. Initially, struct segments of unit length, but not to record lengths, the sidelength of the altar was doubled, however the plague as it is equivalent to leaving the compass uncollapsed). did not cease – it even spread further. It was later realized the omen was interpreted incorrectly. The volume of the We proceed to demonstrate a few of the sim- cube had to be doubled, not its sidelength. Such an attempt, plest and most useful constructions that often come up. would involve the construction of the ratios between the sides of the older altar and the doubled one, namely 3 2 • Given a positive integer n, we can always construct the . Namely, the construction of a line segment with this giv- segment of length n by imposing the unit segment n en length is famously referred as the Delian problem, after times (see Figure 1). Delos where the mystical oracle of Apollo was€ situated. • Actually, given any positive rational number p/q we can construct a segment with the prescribed length as There are a large supply of other ob jects one shown below (see Figure 2). might find difficult to construct with straight edge and • Given a segment of length x, we can find a segment compass. Two famous ones are “trisecting an angle” (given with length x as demonstrated below. Notice, the De- an arbitrary angle, construct angles that split it in three lian problem lies in the difficulty of generalizing this to equal parts) and “squaring the circle” (given a circle, con- cube roots (see Figure 3). struct a square with the same ares). Both of these are € impossible to achieve using straight edge and compass. For a long time scholars have been aware of the afore- We need to make the rules more precise since the al- mentioned constructions. There were many figures we lowed moves greatly impacts our constructive powers. could construct, however nobody knew how to prove cer-

Spring 2009 27 Delian Problem

tain figures are not constructible. There was the call for a more elaborate tool, a concept that would break tradition. How do you show something is impossible? The fact it is possible to prove such a statement took people a while 2. Algebra to wrap their minds around. One simple trick is to establish an invariant or indicator. For example, we are presented with Evariste Galois (1811-1832) was a brilliant young a very complicated object – a figure. We would like to asso- mathematician and a fierce political proponent. Today he is ciate a much simpler ob ject with it, for example a number. best known for his work on Galois groups in the theory of Then we study the way this number variesW asH YweC AextendN’T WE the. . . ? fields. While studying field extensions, he 3was also able to figure. If we can conclude that some numbers are unattain- come to a simple solution of the Delian and other similar

Tablehink ino suchf one ’as fashion,mathem thenatica lweed uwouldcation .knowFor tthathe m theost ofiguresf it, w e sproblems.tudy certa BeforeiWn HseYtsC oAwefNn’ TucanmWbE describeer. .s. ?and o thepera hearttions of his method, 3 on tcorrespondinghem. By the fi ftoth othemr sixt hwouldgrade ,notwe belea rconstructiblen all about th eeither.ration alwenu mneedbers toQ. introduceA bit late whatr, we ean fieldcoun tis.er π and e whThisich isdo preciselynot fit t hthee o pointld sch ofem viewe. Th weeinyk areaorfe ogoingbnoet’hs mitorra attakethieomn ainalt iathecnadl efdorumcatpiaornt. oFfotrhtehbeigmgoesrtsoeft iot,f wreealstnuudmybceerrstaRin. sets of numbers and operations Laterestr, w eofc otheme article.to real ize one caonnnotthecmom. pBuytethtehefifstqhuoaWrHesYirxotCohAtgNor’fTa-d1We .,EwT.eh. .el?ena,rThinkwneaallr ae ofbfo urc one’stetdhteo ra mathematicaletxitoennadl noumr hboe ris education.zQon. A bit l3a Forter , thewe e ncounter π and to the set of complex numbers Ce. which do not fit the old scheme. mostThey ofar eit,b weoth studyirrati ocertainnal and setsfor mofp numbersart of th eandbig operationsger set of r eal numbers R. 2 In reality, all three sAetTsAQNA, RS ,ACTAloNoAkSOalVikeW. HTYhCeAyNc’oTnWtaEin. .0. ?and 1. Addition, negation, multiplication an3d Think of one’s matheLmaatetric,awl eedcuomcaetitoon.reFaolirzethoenme ocastnnoofontit c,them.owmepsut tuByedty htheceesr qtfifthauianre sorertos sixthootfonfu -grade,m1.bTerhs eweann,d wlearnoepaerre aalltfioo raboutncsed to theext end our horizon division by non-zero values are altlowtehlle-dseefitnoefdc.oFmuprltehxernmuomrbe,ermsuClt.iplication is distributive over addition. Despite Thinkoonf tohneem’s. mBayththeemfiafttihcaolredsiuxctahtigorna.deF,owr ethleamrnoastlloafbiot,utwtehsetrationaluradtyiocnearlt numbersaniunmsbetesrso fQ n .. uAm bitbiet rlater,slaatnerd ,weowpe eencountererantcioounnster π andand e which the fact all t1hree sets va1ry enormouIsnlyreina th1eir size, th1ey beQar RtheCsame stricture – what we call a field. These on them. eBwyhtihceh fidfothnort sfiitxtthheg...roaldes,cwheemleeal.irtnyT,ahlaellylabtahoreruetbetohsteehtsriarrtai,otnioa,nlandoluloam onnotbkdeafr ofitslirkQm ethe..pATa oldrbhtietoy fscheme.lacttohenert,abwiinge gTheye0enracsnoeud tareno1tfe. rbothrAeπadladn iirrationaldtuimonb,ernsegR aand.tio nform, mu ltiplication and objects together with a few moredifvoirsmionthbeyfnroonnt-zleinroe voaf l(uaebssatraecatl)l awleglel-bdreafi. nTedh.eyFuarethseormimopreo,rmtaunlttinpoliwcatdiaoynsisondeistributive over addition. Despite 2 e which dLoanteort, fiwtetchoemoeldtAosTcArheNeaAmliSzeeA. ToTAnhNeeAcySaOanVrneotboctohmipruratteiothnealsaqnudarfeorpartmootpao ofrft- 1othe.f tTh hebiggerenb,igwg esetrasree toffo frrealcreeda ltnumbersonuemxtbeenrds oR u ..r Later,horiz owen come can hardly imagine any branch otfhemfaatchtemalal ttihcrsewe istehtosuvtatrhyemen.ormously in their size, they bear the same stricture – what we call a field. These Later, wetocotmhe steotroefalciozemopnleexcnaunmnobtecrsomCp. ute the square root of -1. Then, we are forced to extend our horizon There exists a notion called field extenWsiHoYn.CWANe’TknWowE .t.h.a?t Q Rto realizeC. In oneothe rcannotwords ,computethe ratio ntheal n squareum3 ber sroot of -1. Then, to the set ofIcnomrepalleitxy,nuamll btherrseeCs.oebtsjeQct,sRto, Cgetlhooerk wailtikhea. fTewhemy ocroe⊂nftoarinm⊂0thaenfdro1n.t lAindedoitfio(nab, snteragcatt)ioanl,gemburalt.ipTlihceaytioanreasnodimportant nowadays one are contained i1n the real1numbers, which in1turn are1contained in thwee cW oareHmYp forcedlCeAxNn’Tum WtobEe extendr.s... ?If on oure fi ehorizonld is con tota ithened set of complex 3 2 division by non-zeArToAvNa...AluSceaAsnTaAhreNarAadlSllOywVeimll-adgeifinneeadn.yFburathnecrhmoofrme,amthuelmtipaltiiccastiwointhisoudtisthriebmut.ive over addition. Despite in Iannortehaelirt,y,waellcatlhlrteheesleattsteQr,aR(,fiCnelldo)okexatelinksei.onTohfeythceofnotraminer0. aFnodr e1x.amApdldeitCionis, annegeaxttieonns,iomnuoltfipRlicaantdiosnoains dR Think of otnhee’sfamctatahlelmtharteiceasleetdsuvcaartyioTennh.oeFrrmoeroextuhisseltysmianosnttohtoeifiorints,iczwaele,letsdhtuefiydeylbdnumberseceaexrteatnhinseioss e na t m.s We.ofsetnrkuincmtoubwreertsh–aatwndQhaotpweRreatciaoClnl.saInfieoltdh. eTr hweosreds, the rational numbers doivf iQsi.onIfbEy nisona-nzeerxotevnasluioens aorfeFa,lltwheelql-udeesfitnioend.wFeuarrteheirnmteorrees,temduilsti“pHlicoawtimonucishdbisigtrgiebruitsivEe otvhearnaFdd?⊂i”tioWn⊂.haDtedspoitwee on them. By tohbejeficfttshtorgestihxtehr wgrTiathdiean,kwfaeroeweflecomaonornenrt’esaaimfnloleraadmtbhionetumhtehatetfhriceoraenraltaletlndionunemcaaobltfineo(ruansmb,. swb FterhorairschtQh)i.enalAmgtueobbrsinrtaaol.arftTeiethcr,,oewynweteaasrietnenuescdoyuiinmncettperhtroearπitcnaoanmsntedptnsloeowxfannduuammybsbeoernrsse. aInf donoepfierealdtioisnscontained the faFctigalul rethre1e1. seCtos nvsatrryu1ecntionrgmoaussdelyeggmirneetnht1eiorfsiinzet,egthe1reylebnegatrht.he same stricture – what we call a field. These e whimcheadnobnyot“bfiitggthere”o?ldWsecohrnefmethre.etmoT....thihnBeeyyanatorhetehbfieorft,ohfwEoeirrrcoasavitlxielotrthnhFagel,rlaoandfttdete,enrfwodareWmel(neHfioapYetral endCrd)tAa[lNoeElxf’aTtt:behWnFoInesuE]i,bt oreality,.iwntg.h.hgo?eeifcrrhtasht eieallistofntoo hafthreerelmrnedeauirmlm. nsetsebFuneomsrrisob enQe x r .aos, mfAREp. bl e,ia tsCla tilooksera,nw alike.exetnecn osTheyuionnte3or fπRaanndd so is R objects tocgaenthhearrdwliythimaafgeiwnemaonrye bfnorramncthhoeffrmoantthleinmeaotifc(sawbsitthraocutt) tahlegmeb.ra. They are so important nowadays one Laterv,ewcteocrosmpaecteoorveearliFze. oWne wcaihnllincnhoottdocoftorQmnyo.pttuoIffitdeEtettshihcsereiabsonqelduetaxhsrticesehnreinsomiofetun.loloTffbh-uF1et.y, eTtalhahrebeeonqbr,uoaewttshetioiarnrreaawtfnioeorenacxareaedlmaitnnpotdleeer.xfeotTsretmhnedinpkioasuro“trfHohtofhowertihzcmeonumbciphglgebexirggseetr oisf EreatlhannumFb?e”rsWRh.at do we can hardly imTahgeirneeeaxnisytsbarannocthioonf cmaalltehdemfiealdticesxtwenitshiount. tWhemk.now thcontainat Q R0 andC. 1.I nAddition,other wo negation,rds, the r amultiplicationtional number sand divi- to theplsaenteo. fWcoemcpalelxitnaumpTlbhaeinnLreskabtCoeef.rc,oawnuesem’esceiomtamnaisetbhttyweom“orbaeditagiilmcgizaeeelrn”eos?dinoueWncacealtniraoensnfoe.atrFsctoporamtcthpeheuewtdmeheegtorhsete⊂ebosoyqffuiot⊂Ean,reweovedreoismrotutFedno,ysfoio-cf1tnee.rnwtTaedihneernensofe,etrwesdetoof[aErtnheu:emfForbre]c,eaerwldshationchdexiostpetenhrdeatodiuoimnrsehnosriioznonof E as a TherFeaiegrxeuiscretosnat1a.ninoetdioninctahlleedrefiael lnduemxbteenrsi,ownh. iWcheiWnknHtouYwrnCtAhaNrae’tTcQoWnEtaR.i.nsion.e?dC inby. tI hnnon-zeroeoctohmerpwleo xrvaluesdnsu,mthb eearerrsa.t iIallofno anwell-defined.lenfiuemldbeisrscont a3Furthermore,ined Inlinreea.liAtyn,dalilndtheerdee, [sCeto:snRsQtt]or,=uRtch,2teCinwsgelhotveaoerokcesftetoacghrolmiemkseepgpn.aetlnceToexfrhonaievutnyeotmrrecsbgFoeon.rftsaWlteiChnee.gw0tchoial.mlnnpdolet1x.trnAyumdtodbidetireossnca,rrinebe1gtahtniisdoniin,. mfOuulnlltbtihpuelticoealtathibeoornrhaaatnenddo,nthane example. Think of the complex are containeadnionthtohenre,trwheaemlcna.ulBlmtybhethrslea, tfiwtfeht1rhicahor(ifinsiextlduth)rngexraatrednecs,oiwonnetaloienfaertdhneian⊂lflotrahmbemultiplicationo⊂ecuro.tmFtpholerexreaxntauiommn pisballe distributivernsCu.mIisfboearnseeQfixe. toverleAdnsibsio ictnaddition.olnaotfeaRrin, eawdn eDespitedensocoiusnR ttheer π factand divisiroenalbnyunmonb-ezresraorveaolufeiqsnfianreitaeIllndwergeerlale-leditneoyfiv,neareldlt.htehFrureereathlsser–tmswoQer,ecR,am,nCnuoltltoipfiolknicdatlaikofienn.istTedhniseutymricbbouentrtivaoeifnogve0nerearnaadtdo1irt.sioaAns.dadDbitoeisvopen.it,eInetghaetion, multiplication and in anotheorf, Qwe. IcfaelEl wtihsheiacnlhatedtxeotrennaost(iofipfineltladontf)heFe.ex,oWtltedhenescicqaohlunleemiosttfeia.othnTpelwhafenoyeramarbereeerci.nabtuFoesotrehersietallirexridasa mthreetitsiwpo“nloeHadlC osetsiwamnisemdna ufsvarynoicorhenmxbat liepenormouslygnagsreitroanoisfpoEtfahcRetehb awiningh dgetheirFersre?o”sbeiy stWsize,Rohnfaer tetheydadilomnweu nebearmsiboenr sthewRe . refer to the real the farcetstaollf ttThhhriseineakrsteoitcfsloev,naewr’yse mewndiaoilvtlrhimuseismoeunafistblnliyiycintainenl.oentAxdh-tunzeeecdinrasotisinioivodznanesl.,eu,dteFth,shoe[arCytrtehb:isaeRlaelm]xrw=toteshnl2tles-owdisofeahnfiimetsnr,eoewdfste.tfihrneFsitciugutreudtnryhdeercrg–aemrtrweotoearhr.siean,otmsfwetutehsltecoipacflolinmcuaapmtfiiloebexnledrni.ssuTadmnihsbdetersoriebpsueartraievtei1oonavsnedr aid. dOitniotnh.eDoetshpeirtehand, the of Q. If EFmiiegsauanrenbeyxL1“ta.ebtneigsrig,oewnre”o?cf oFWm,eetrhteoefeqrreuateolsitztiehoneondweeegcraerneenooinfttEecroeomsvtpeerudtFeis,t“hoHfetoesamesnwqudmaeru nestricturecohrtoeboditg[Eogfe r:–-1 Fiwhats.],ETwhteh hweniac,hn wcallFies?at” hraee Wfield.fdohirmactee Thesedndsotiowneex oobjectstfeEndaosu atogetherr horizon objects tFooigneeltdhseramlwl.oiwBthytahthefeCawficoftnmttihsohotnerouerfocafstociixatrnmtdgarhjeloaltagihntlr1sehaiennrdgfuegremmeo,nebwsnlteeettrmlsisloneefvaenaritonernsfyt.oea(fgealTlneibnrahosfibretlmrenoarniuotacgteutt)itshdohla.nyeelgagirlneeabnettruihoamevn.iebarTrleshrtnisehzuyema,arbterhearaelsyovQ–bei.mrweyaAeprcocbotrahnitntaevnnlesaotnatiemnefirone,nwtwdsafitedareaielcfidyntnscufioortuernnen–mtuewarmnhπbyaeatrnwodef gceanlleraatfioerlsd.asTahbeoseve. In the mean by v“ebcitgogrers”pt?aocWteheqoevrseeerfteFro.ftocWotmehewpldielpelxgn/rneoueqtmtorbfyeErtsooCvd.erscFri,boeftehnisdinenfoutleldbu[Et e:laFb]o, rwahteicohnisantheexdamimpelne.sioTnhionfkEofatshae complex WHY CAN’T WE . . . ? 3 can hraeradsleoynwism.hiaHcghoinwdeeovanenroy,tibtfirtiasntohacbhevjoecrlfdtymssrtaceorhtseghtesermtitmcheotae.fivtrtTiechwhsioisetnwyhaeirtathrioofceoeluew.bt,oFtwmtohheorermiwerexri.lfalotmruimospneleatfi,hlnetaihntferwithdosenfqxotutr lemaainr esefewpioraoonrftos (,moretaotbfohsftath2rt eaformihcsbati)esgxagft aleethegmrnesboieru otafrontns.losyfTobrhfee efialineeylninatpur eerofmodsvbe oe(abstract)gedirmesepR.o.rtan talgebra.nowad ays one vector spapclaenoev.erWFe.cIWanlleriewtaliailtlypn,loaatnleltrtbyhertceoaeudseestcisrtiQbise, Rttwh,oiCsdilniomofekunllasilboikuneta.lelaTasbhoaerysaptceaocnoetnawianhne0reexaanbmydpo1lne..eATddhimdinietkniosonifo,ntnhwegecaortmieofepnrl,etmxoutlhtieplriecaltion and ThtoerbeLeeaixrtiresart,tsiwoanenacol.otimAonedjtcooainlrleicendagnlfiiz(heaeladorddneilxenytgce)ianFmn√siaineo2glodnitns.oecWaoQalmlen,opywkunebttoreohwabetnthtcahehciantsotiqoafQunmnareoaewftRhrafioedmeojltodaCiotTheyndi.ficen-sIn1gnwo. toeareiTettldhehmoe eQsournet(n,w √importanttwosh2r.ee)dm.asT,rP.hetlehefoaerrsar cenowadaysateintdoioontnaoeal telnhxnuetume nmnonebedbweero rsficanuseralrdhe hardlyoirasizvoenr yimagineconve nanyient field for many plane. Wleincea.llAitndaivipnilsdaieonenedb,by[eCcnao:unRs-ez] e=itro2isvwatwlhuoeersedaitmrheeanglsleiwonnearllal-dtaoesrfisanoesfdp.tahcFeeucrwothmeprmelexobryne,uomneuTbledhtriiimspnlkaeicrnoaesfti1oioonnannewi’dsedimr.iesaOftterhnribetutmohtaietvtheoicetoahrvleeeerardlhauadcndadtiti,oiotnhn.e. FDoerstphiteemost of it, we study certain sets of numbers and operations not taolltrhaetisoentaolsf wcoimthpolenxe neuxmtrbaeerlsemCp1.e/nqt √2. In order tfioelpdr⊂esxetrevn⊂esitobranchhne stru coftu rmathematicse of a fiQeld wR withoute neCed t othem.make sure are containedrienalthneumretbhaelernsfuaacqmrtebaeolrTflsith,nhewfirrehnreiectaeshxesidtsoinstensgsvtr.auaerHenynootaewvinreoeeornvrcetmcohra,neoltilurtaesedilinayslesadin–vintwehrteyehicreraescnsiontzmreoi,tcp.tfiliheWvnxeedyenoaukbnmnefiaonbtrwioettroeths.h.neaFuItsfomarombneeeexrasfiotmefrlipdgcleteiuns,re.tcerhoaI–entotwsoarqtsihnuhaeatsdreawwbreoorcvodaetsl.,loItfanh2fietherleadts.iofTanmhaelosnueuslmybbeeresn proved Q lianded. inAgn,Idnsuinrbedtaerlaeidtcyt,,in[Cagl,l: mtRhu]rl=eteip2slyewtinshgeQrae,nRtdh,eCdgivleoindoeikrnagatloaikrrse.owfTetlhl-edycefiocmnonepdtlaeoixnnn0eulmeamnbdenr1st.saoroAennd1tdthhiaetenmiodnn.ei,Bw. nyO⊂efigtnehaldtehi⊂,ofienhfto,ehntmhcoeurrlswthiixepatlntihcdaakg,tertiaohadnelel,awnde learn all about the rational numbers . A bit later, we encounter π and in another, wreesctalolfththoeibsljaeatcrttesircatlraeoe,g(cwefioteehnldtewar)tiiolnwelxebiudtesheeininarsfiritfnoaehntiwteioeornmfeaaxotllth.reenAnuffsdomoiojrrobmniesner,tirsnht.,hgewFaf(htoraoridicnsdhetexinixlanigtnme)tenpu√oslreifn2o(CnatasorbieosQtfcar,ofianwnnctetiat)xeoitnabedeltngdeasegiibinronrenaet.honefTeRwhcoemfiaynepalddrleesxdoseonniusoimtRebpdeorQst.(a√nIft2o)n.noewPfialedealasdyesinsoocntoenthaiennedew field is rpeaolssndiubivmleisbcieoornmsbbayirnenaotnfio-inzesfirnwoiitvteahldutehegsersaeereeoopavelelrrawteiholeln-drseiafinlncsleu–dd.winFegucrratanhtneioromntaofilrnned,uma ubfilentrisptelaicnadutmi√eobn2we.ihrsTiocdhfisigstderfionibenlurdoattinvofiaertsouvatrhesarelalaybodloedvxiets.ecionhInnde.mstDhteh.eesTphiteey are both irrational and form part of the bigger set of real numbers R. of Q. If E is an eFxiteeldncsiaoannllhooawfriFdntlh,yaetnihmoaetcahtqgienuoirnoe,stewtoiaefolnnlacyradwabljloteriitaonahnriecneahgllisanotewtfteleimrtrmehasaettoneh(ndtfieseme.ilsedaxT“t)tiHhrceaesoxwtwerelainemt mshiuoioencunhatlobt√fnihgut2eThereghm.eerbI.nfeiosrros Emr existsaderetrehr.ataFno ovF arpe r?ee notiony”xsaecWmrovnphevla etthn calledCedieosnisttwrauefi nce fieldtleudxretfoe onr extension.fsmiaoanfineoylfdRw ea Wenndee sdo tiso Rmake sure rreastiotfnhaethlsfiasacnatrdatiliclslteoh,frwedeegwsreietlesl utvwsaeroyfi(neniototeprtmehpxoet/ue2nqslsiynioin√ns2,t)ht.heiaArtnsiiszeexe,xtetnhenseiysoionbneosafrodftehfigenreisteaemtdweLogsarittesreicr.,atulwlreeedc–aomwquheatdtorwarteiaccliaezlxelteaonnfiseieocldna..nnTItohtecseompute the square root of -1. Then, we are forced to extend our horizon mean by “bigrgeears”o?nsW. eHroeTwfeherevoretfeor,Qetixh.tiesIitfsdsEeaagdrvindeseeoirnatyonigof,rneeEsxsuctabreoilntvclrestadiriocvnFtfieien,olgdnfo,feFetmext,ntouteohlndt.eisepinFqoloyuntrie.nesedgWtxio[aeEnmnknowkdw:pnFloedewi,]av, trthatitwhedheihiannitsgct qheQuarireasesrtetweRhdreolilods-tdiCm“eoH.fiefInnoI2nwes iotherdhomatnohsunoecfrfahe words,mElwebomiagruesdgsnsleayt,r sthetibhsoeenEe rationalnrtathtpheiroaonveaeFwl dnumbersn?fi”uemWldb,herhaset ndcoe wwee take all FioebldjesctaslltoowgetthheeracwtiitohnaoQfeaw√djmoionriengforWemlHemYtheCenAtfNsr.o’TnTtWhlWiEenHe.rY.ao.t?CfiAo(anNba’TsltnrW√auEcmt).b. ea. ?rlgseabrretao.atThvhe⊂ersyetac⊂rooenfvscoeonmimiepnplteoxrfitenalundmt fnboore3wrmsadCaan.ys one 3 vectotrusrpnascoeuotvowebreeFcai.rnrWacrtoeeinowcnsotairlnllu.mtcnaAetoinadtnejtd(orbiyniyn2itpn)“otogbhasdise(sgeaiagrbsdlecldaerni”lbcnu?nogemum)Wtmbhb√ebirsn2ersareitonstfoei,fofrtQwunhtlhs,eloiwcbftoheuihtrteohminbedtltatehaaug+ebirnsoenbeeraaoot2prnefeeaErwocsanoatfinoiaoevtanalnedirsdnedxFiebnead,cnvmlioaunpfrtdtyelteidhno.egvQTdecr(erhoa√nimtnohi2oktpe)en.lodreaaPxfl[ttlEnniehouuaen:msmaecFblbosn]eem.,orrswStpse.ilahmentIixcfhidlheoa√nrinlsey2e,t.wfiheTefildhedilisdsmfiiceseonlndstionanianteoudfraEllyasexatends the reasocFnasin.ghHuaorerwdelyv2e.irm, aitgiinseaavneyrybrraenstcrphicotfivme aotnheemtoaot.icFsowritehxoaumtptlhe,emth.earesqu containedare rootIno f inr2e ah theliatsy ,f realaaWmllHo tuYh numbers,srCleyAeNbse’eTetnsW Qp whichEr,o.R.v.,e?Cd inlo turnok a li areke. conThe-y contain 0 and 1. Additio3n, negation, multiplication and planea.dWjoiencinagll aitnya iprlriaanCnteioaonnbosaetclvhraesueucqrctsu,toeaiwrnrietesgpcirsaaoacltotslewiteotoghn(vmneadeolrilesmtanFattetn.hnedoarsWiftoiaesrnoa(foaw√fitflaieiloadldpsnen)egaaorrletfesexleptcetrteantynwcgsestqotiohouw(na.dnhreoeoestrfcert.ihgbbe.ye42foto)hnirniwemso√deiunri2m.ldf)ueF.nloelAsrabinodeunxtetaxowemtleapnbrlqseoiufroeCanrtdeitorsoafotandtincheageenrxreteeexantlawsmiioopnilse.o.cfTaRlhleiadnnkadoqsfuotahidserRactoimc pexletxension. It ThtionkbeofirornTaneth’iTosoethrnmeianalel.ktxhrAioaseftdmsijoooananitenain’cilsonastglmwioe(aidnattuhdhccdeaoamilntleiaegodt)neicfi√.xaetF2lrldoaetrdoeeuxtlQehtcemae,ntewmsinoieontnso.t.bFot2Woaf.rieinIttn,khanweonoremewdwseottrsuhfitdtaeoyltdfpQcidrettainede,restnweaoRreitnvese dtsinuetChQtd stheeyi.(vo√sIicftsne2 rincomplexro)uout.nctamthPibuneblyreeseanrwessoteosnf arnnumbers.-onadzofdsefit,rneeotulphtdmvhearebwlraueane treIftiesoniwo neaonenesrnfiadedellat ndofieldloulpimwserebal ekltis-riedo scontainedensfiusnreed. Fu rinth ermore, multiplication is distributive over addition. Despite line. SAinmdilainrdlye,eadd, j[oCin:ionRfg]Q=a.nI2fpirwlErahaniteesir.oeanWntahtleuexcrcutgnaeebslnnleoseiiurtooatnatowotoperfwlsacFoanu,fneltdcbhoelnceqcsatoudrmesusetcptotiloteQaxnis(cnw√utuebw2mi)aocbradeesrxiimasntletlaenenrrsesueiiosm1ontneba,adenolrdirsassio“.fnaHOtehosnpowefatfcmohdereumgwcorhtheahe+beri3rebg.√ghbeaFy2rnoaodirssn,eeEatxhdaetnimhmdapenblnevsF,aio?rn”y owWveehrraettfhederortaowteitohnealrse.aSl imilarly, not 3all rFaiatgdioudnreianlgs,2w.suitbhtroancetinexgt,rma ueletimpleynitng√a2n. dIndiov1ri/dd3einr gtoa2rp/er3ewserllv-edetfihanother,ne⊂esdtroun⊂cteu tlwereheme oecallffanctats fitheaoelnld tlatterhwreeennese weatd (field)sfitevoladrm,y haextensionekenneocsreumrwoeeust laofykie ntheatlhl eformer.ir size, they bear the same stricture – what we call a field. These onretahlenmQu.(m√Bbay2er)erotsinhcsaeottrnhfihetefeaotmihsfne.ieotnBdrofiysfinCniaixttohltelnhensdetfilgueirrfnmguteaerhcabde.tleoiAn,rrosugwnvsmodieaxfrbltitsenhteheahdrgergsemn,rferoawdeaerdn,alhmeltl[i,sCcaoahwb–f:+oeirwRnualbet]t2Ttai=utcohrharnine2na+anklrawlaoclrohet2eaifneofibcrgononetnanudhewtlt.h’anshetiuehnmfirgmeneeadibrntaaehei,ntrrebinsamo,tutQcnhoamaea.rtlbisrcAeneoarurmflbamoetipttfhdbiloeugleanxecrctansoenlterQmsuir,.oam.pwnStlAb.oeerxvFesbrenosnira.trcausolImtaufhatbnoedeteonjrmerov,ersiewonfi.πasierntIlaenegdonns1ftdichsmiaotecnu,aoydnwntebtie.aresiOtnπuendadnythdceerottahienrsheatsndo,f tnhuembers and operations adding, spubostsriabclteinmcgoe,mamnbiubnlyatitp“iolbyniigsnggweariat”dnh?jpdotWihndeiesnvegrideoifapnenregyrtaaoitriertoahnwtesieodilnle-acgdlreuesfiedqniunoeafgdrEreoanrtooivoeoelnteram(Flnen,onuotmsftbehoonenabrtsdjteoheacnfentosdantte√podweg2re[.ffiEtehTecle:thdrFi,sswq]h,fiuietewanhlrdcheaeincefhwae.gtewiu.str4mta)hklolewyredoaeiulfxmloltdreemnnldestiasohdtnehtfoeorfoanEtqaluisnaeadroaft(icabesxttreancsti)oanl.gebra. They are so important nowadays one e rwehsticohpfedtrhofoiinsnrmoeatrnewtdfioihctiilhnceteh,hsrwe,dqewoulewdennioclscatlcelhufil.estrFemhetoaefiherl.nelenaiTxtoutealthmdemerbxyspecatlrahees(nre,fiemsaieifrboleewdno.)oestTh,fetthiaxinhoerktfiarnyeatnQtaisihitrisoeoewnenmdbeaxe.lotmgeftaBrnhanteysyhdeioertfiorfhnorvfaesosrettrmirofiomefftntnhepfihalreanal.orirraFortrgFeneeaosddilfr txsexampleeofteto–ghxQreawmeg(meber√.aipcg2dalg)ern e,tarCnwnos deftieisstlt fiehanaonaenfdr nnbextensionreaeixfaguatfilglrelnetnnahirstubeiesmorenutbuto eoofmrtfQhsb Rr(eRe√ara. al2andnot,nfdi√ougn sm3esooa)nb.lei esnrisrau sR tmoRrb. seofrass Qab. oIfAv eEb. iItnlattheer, we encounter π and rationalsveacntdorisspoafcdeeogvreSerimFtwi.loaWr(lneyo,wtaidtllhjoneion2tinitngry√atn2o)i.drreAastcniroienbxaetletnchusisiboiennrofoufoldtl√cebwagunroteuheeldaltarwbdleoolayrdiasitmtceoaolgalneindcaeunabaeinqcxuyaaembxdrtrpaelnteics.chioTenoxh,ftienmonkrasoitohnfneet.mhoeIafttcdiocemsgwrpelietxh3o.utFothr eemxa. mple, Later,pwosescioboflmeLQeca.ottemIorf,brEiewnaeialsitcziaoenmnosenexwettoiectnahrnseitnaoholneitzsoeecfooFmnpe,p3rtucahattenionqtnhousteiscnsetoqciomlwuunahpdrwiuiecntehgreoadrtorahoetteinioonsofnqtt-eau1rfila.etnrsTeuttemhrdoebnoiisoest,rl d“wsanoHfaes no-cextensiona1hwdr.eemTf2oehu.recchnTe,1dhbw ieof3tsgeyogfi aeFaer rxl,ede ti2thesefnbo3nEaordtc questionuhteohdruairtlnrlohyaFtoeeirx?oxit”nz teweoeanWnldd aharesnoautdthr interestedfedohoromrwizepoanrt oisf “Howthe b igger set of real numbers R. FiSeuldpspoaslleowF tishea afipcetlaidon,neE.orfWaeasfietdnjcoiotafielnlteihinQxtisgt(aea√nerpls2teli√i)aomcnilesen,ottbfwhseie.tcsawaTeuntihsldloeefuLirastaelalitsfiinfiontunwimiattoelebndeexurixmstmteeonbnfsesstirioihosnenasfa,roletfrham√Easa.tavaTie+ssrhpybeea2xnctoe/e[nLw+vseih:ocneF2nires]e/n=obtfyw[fifiLheonlen:idrteEefd]oad[irE,embgm,e:rcneaFesan].iry.oenTrawhtiesiorneafelsr. tSoevtherearleaadljoinings may be to thersaettioonmfaectlasoFntmatuinhgbprdeynulesresix“seobtnuoigu2otfgm.fwdecebCerog”eocmr?ranespnesWCtlectre.wxouncorntesuit(fnernmgruobctateotrQshstehe(gCem2.2dei)engatr√seoLeaf2al)rolt.afentrAuEi,omnwnobaevelxecrtlrsoemnFogsf,eittoohhtnfo.eteorfnoferadmleiezgnaero+eoteenbdtew[c2Eoaani:snFaTocta]ha,lcenlweorddemhbieapcxvhuaqisturitesyastdtoahrvaenetroisdcqttiihmuoeeanxertrnecasnatirlosioloineondtnao.filofsfeE.Ilt-dS1ai.emsxTitalehanerslnyi,o, nw.eWareekfonrocwedthtaoteQxtendRourCh.oIrnizotnher words, the rational numbers reasodnesm. Honoswtreavteers, ittheilsindae.gvrAeerenydarceiFtnssitderileindecsdtpai,evapr[elCflrooewr:dnmRiecte]thtdao=eboinla.2ecsFwteioqoharnuyereueoxnfpactemoha.nedpFjglmoeoei,runtlietnhrxiegpaatlmseoeqrlpeuslxmamucheot,refeinfttrshwsoieo. oebiggerncttToa(omhakfdep2j lQorisehaix nEatwi snonthaneungfama)m.lab yonFeur fi?”ssmrl aysbWhattreeberens1elna rdongpeder aoweitv.oveOed Qmeanrny(√tch2oe)n byovaten hnd“bigger”?eiertnhhteanfinefdul,d rWethfoer r mtoaQny(√2, √⊂3). ⊂ In trueranlistvyoe,cutatoalwlrdInejtsohpcirraneecniaenelcigsotoeyvnat,esnratyrQlFlui,r.ctRrthWa,rQteCeie(o√wnlsoa2eiollt)lkssanqQsaoulta,iaklRtrlere,n.yCurtoTmolohobtdeoteeyok(rsnsctcaoohroltifienbktteshae.eatitnhtfToiso0fhrfmiecanyaonamfdcpu+opel1lnlrbe.f√tbexauc2Aintaduess0dmlqaiautbaiaaneornrdnresad,1teCabne.rg.evoe.gAanac4rdoty)aidnonowitntvaeoi,eoixunrnmaletd,mduhnleplitenliragepad.atltihtciTtioeaohntnriaean,olaknsmql.unouaSafunlitdmtdihrpibaelaiteccircaolsyt,me,iwoxpnthleeinacxhnsidoin.turn are contained in the complex numbers. If one field is contained to be irrational. Adjoirneianlgnu(amrdebadseinorsng)sa.r√eSH2uoopftwpoienoQvfiseen,riw,Fteitidsoiesbagtarafieievneledora,yvnEererewatshtfireinecrlitdteieavdleesexon–tnoereferwetneestdioocQoan to.n(on√F fotheo2ittr)fi.ae ndegreexnPdadlemaLapsfielane n,ofifitotne thiEetne eoverutshmeqexubtn aeFren ,ew sooftenirfofiongeoeltodnf oedenotediErfsa2.toThrashsean fs[Ea[am L:b Foo: u],vFs ewhichl.]y=Ibne[tLeh nise: pEr]o[Eved: F ]. This divisioandbjoyipnnlioandngiv-Seza.iisemnirWoyoinleavirbacrlylyau,tlneliaosidntaj-araozelienpsariqolnaluvgnwaaerealeulnble-erdsiocreaorfiuratnets(eiaondlnio.ltatwFlisteuchlrtluat-wbdhtoeofirdfrImnnoiameodrtpe.eneaw,Fsrlmoifutoeuryucntl,ltdahtailesplrqaellmuaisctdaoharrteteiso,peenma.asgcieu.sectlu4sdtwib)ipQshitlwcei,rcroiReaibeunx,tultbiCdtoaeyinvllsoeiiastnoohdekndve,iedrtsaro,tilmrairwkiadbeodnu.qncistuetaiTiavolooldenhnfrteo.awhdyvtDeeicgcrelorrasaenptfdxeteiadter3eriint.taioso0Fin(otofi.hnareen.Dledrxe)esa1apem.lixtpteAelend,sdiiotniono,f tnheegaftoiromn,erm. uFlotripelixcaamtiopnle aCndis an extension of R and so is R not all rationals√3withroenste oefxtorhaibseealerirmtricaedltneit,omnw√oaen2l.s.wtArIilnadltjuoeosriednt3eihfin.rengTtidot(heagpederdxeSitesnoee1gnarl/)cvsu3ite√otsnti2ihosn,etn2oat/sh3tQpartherut,ecwdits ieudimensioncertoxeabtbeotlanfeisaniwofianey snofldeuowf pwEofi fieasnenil mtadee euvectorddleteintgpoorletme ed.spaceaxkQtee(ns√su i2overroe)n. (Pa Flde .ja oWesienin nwillogt)e. tnothe ntryew field is the facStimailllaitnrhtleyhr.,eQeeAa(fdnasejcdot2tis)inanvildsnlaegrtteyhda√ere,nesn[eeCxiortsrre:moatfRtsoia]ouvln=slalanry2lyuicmwneunbhbtoeherremserirootofhustiteszhlweyged,eoiiftnvunoheilrsdertmiyaholteneaboiarrbe+dsaysrbiotzn2foteoht,nahet-eh+zcseuaccrybomo2mibcevpeaesaltxeuwrrtxeihetcshnteauersreirmoeseanba,m–,oelblfro,ewwsrQchseaaot.alrrnlteI-ieecfdw1treEouaefiafrtnniecdsoeadend–lagla.inwr.laFesehO.eufixarnSet3ttel.hdtnwvhe.seFereirTomacornhlaotealrholsxedefe,ajrFomfihi,nupaetilnnhdtedig.,ep,slqTitmcuhhaeaetsysitoeiboneniswdeisatreibiunttievreesotveedr iasd“dHitoiown.mDuecshpbitiegger is E than F ?” What do we adding,3subtracting, mulFtiipellyndiosntgaalallolnwrdatdhiiovenidalicsntgiwonaitrhoefownaed1eljl-o3edxinetfiirnang2eedl3eelmoenmenetnlet√ms.2e.nTtIshneoonrradttehiroentnaoelwpnruefimseelbrdve,erhstehanerceestarwuvecettruayrkeecooanflvlaenfiieelndt wfielndefeodr tmo amnayke sure Q(√2) isptherefosremt oefdailnl√nseuxqmubeenrcse.ofFtohreefxoarmpale+, ibf2w/e t+akc2e Q/ wwehmeraeyafi,tobr,sct describeeanrelarramgteieo tanthisona3Qlbs y(.in√S“ 2ebfull)vigeagrn aebutdlr”at?dh jeelaborateWoninefuinrregtfhseermrt atoonyotb Qhanee(√ dexample.e2g,r√ee3)o.f E Thinkover F , often denoted [E : F ], which is the dimension of E as a obpjoescstisbtleogAcreoettmahotlbehnrijnseuwcamptittsoibhoitennoratssg,fewaewtrwiheteehmhortafwohavirendietsfidhepfniornoaiortgpmvef,eidrswtauehtbdgmeirtoeorfnrnaersoecoontiunfvitogneclrhgrlimun,tbdethmatihenohcuefgkerltefg(rfaiarpacobltlstuysniot–niatrnldalwlgaicnlotetahen)cnruoaedfimfnleegn(dlbesadoiebvbtrtrtissdahfit.airenvnaonadTgcrdrythaa)√ertfieyaoe2nla.goiwptrerTrebmolhrnlsvo-aiouedsu.meisfi√Tmfilebynh2lpedeiorindysrnotatnofahtrnonguetteirersnacnloeolsolrmniyiwazmsetaetno,pxrdrtuotashercytnaeosaiydsnbnoslatnbettbehnhsoaeoevrwlenvt.aeihndwIengaysfitasehmloedne,eshterniccteuwree–tawkheaatllwe call a field. These SupposerFeasisonas.fieHldo,wEevaerfi, nititieseaxtveenrsyiornesotfriicttiavnedoLneatfiono.iteofFoe rxtheteexn ascomplexm√iovpnecl3eot,.of rtEThs.eh pplane.eaTscqhSeueoaonrlv e[ueWeLrtroF:io Fcall.tn]Wo=f eit2[wL ah√i :laplanelsEn√f]oa[Etmt o:rbecauseuyFst]l.yoTdbhesei scnitr ipisbr eotwotvheids in full but elaborate on an example. Think of the complex can haprtedhrlefyoDrriemesclteaiadgonnifinhntpeahrsraioedsnbqlayulerembtinmircc.alaeen.g,ciFhwnoeeorafwenmxiylalambtuhrspaelnmefic,anhitfiitocwe√fseemwxtaittetkhohneobesQumijoetncawttshsei,cetmsmohwag.yetitfiihsroeserutxtwtetenihtnlheasmiraogn.efsetwofmQfi(onriet2ef)odaremngdrettheh.eenfrofunrttlhinere toof √Q(a(bst2r,act3)).algebra. They are so important nowadays one rationals and is of degtorebeetwirporao(tsnisoibntlaetlh.ceoAm2dbjionininaitn2iog)n.(s2aAdwnditienhxgt)ehn√es2ieootnopoeQfra,dtweiogenreosebinttawcilnoudaisinncgeawrllaefitdieolandaqdluenanudomrtaebtdiecrQse(xa√tned2n)s.ioP2n.l.eTaIshteisnfioetledtnhaetnuerwallfiyeeldxtiesnds the ThSeureppeIxfoisFsetFsiFdfiaeTalmidhnsfiseoaerntaleifidsoltle,norlxbadwicyt,saetEtltslhhetaeedahnaeFfiocned-tlpiiedtolgeanernexceoxetaeftwalnelacnesdtdissomjionfiiien.enalioWdnanfgeFpixtrkceta,enldentoimhcdswhitaeoaLatntnbrhti.dalsae.lWtfiyawlnQTeliaimpthykeoeanuigeoRnrpixwantotsteenitn(ohaxCmnsani,a.toyuydimensionallnlIQ)tbniosrpuuaoflmncteEhRcbe.ehextrrThptoswealfhCantoaemn s.rnbasiedeaoIo.[snt tLa,ahWh o(tespace:vahmtxeeFhderaceaj]yrotarna=iilcdnwlt soiwherei[ioynLntwrngvdaai):etrs.lnEhe,pniolt ]eauby[hunlEnmettme btr:onefihabeFenterelism]tdco. sandimension.TufoaosfhlreFinsmiu.tmains3byet wrweso dreferimen tosio nal as a space where by one dimension we refer to the real turns out we can constnroutctalQl r(ra√att2iioo)nnaaasllssalawlnnidAtuhtmisotbhnoefierssdepeoxogftirtneahtee,etlfwweomeromeh(nanatvo+et√⊂bpt2√hr.oe2vI⊂2niadsioenadrd√aeennr2do⊂)t.ubogAvphanr⊂rbeyesaexocrtkvvegeenrsottihuhoenedsrotaofrtunidocefintguearlrledsee. totShfwiemaoorifiilysaerltcdloay,lwpleredonvaeeq√dua2tdoirsamtniacoktecxsotuenrnsestriounc.tibItle solving are codnCteamainnorenewdsaaetsirnoecanottcsenho.ssentHttrrhaueoeiacwntldendeavgulelirmne,√pebtiotheiaxnreicsstt,rsseawoainhvnl eincartuhyhpmeirnbeQsdett-riurpcsrit,lcnawtcbiavehlreie(ctwohchnaoeiTeynrthtutaeoupitronireoo.nennedFaxmoirlinserupttlcselthaoxianepaetlnmcea)o?pietmnlixeoWptthe,dneletnichxnsha eirealnloatlshnuveqdemu( cadfilline.baiodinersmejlrecdos.upri .nAndeoslAeIsxionfxenttedgodn o)nindeed,ua.sfienim2osdfienbehg.elamdrdWss, ei.sfne[aCItcfmkoon:nofonRutweas]nilfi=nytyehelb2dardaetw etiwhereQsinhocenporarenloRttv atheheiendegC dgeneratorse.nIenraottohresro wf othrdesc,otmheplreaxtinounmalbneurms abreers1 and i. On the other hand, the adjoining any irrationaadldsiqnuga,truserunbrsotroatuct(tnihwnoeegt,DctamehnlauiactlntoionppfslrtyaoriunbpcgletrmafQen.c(d√t ds2qi)vuaiadsrinaelgle.nagur.em4wb)ewrllso-duoefldfithnledafdoorntmoeaale+mqbue√and2tsraaostniac taehnxedtenbnesvwiaornfiy.eoldv⊂,erhtehn⊂ecerawtieontalkse. aSlilmilarly, in anotlhenerg,tohwiebnmecaaiynrllrobattethieocron,lnawslt.tereAurccdaatljelo(ditfin.heienlIdfgl)aw(teatxdehtrdeainavnseg(io)fitone√lcdo2o)fntateoshrxteeQrtuecf,ncowtsnrimteoaaneopinrb3o.et.ifdanFtTitonhinrihenaetfhtxnohSeareemowrelrepfiauofrllt.etiln odiFtheCounodmanriels bnecomplexpaexolrnat2esemna,edelwxp,nQtlhweui(ncem√C shnumbersimb2oie)sna.rysatoPnusfaltrreReaneaxr saretoaeefrnwnenid nis1octi fisthoandoennsititeoasheg fii.emnR dROnenedegnawrt intheensdefioteofhsl vodothereeicrissotmRh ephand,lreexalns uthe–mwb eercsa.nInfootnfienfideladfiisnictoenntauimnebder of generators as above. In the Similarly, adjoini1ng apnoisrsriabtlieaodncjoaomlincbiuninbgaetairnooIynfosxtFixrwwriaifotthuaioltdfinheaellledsaes,dqboupytaoertrehaetrcioFouonb-tspicl(iannecoxeltutewdtnhiensaigmtonreoa,aftnoiaorFnpoaen,lrefntehuocamfttdbsiqesgurasralerlaeenp3doe..ign√.Ft2so4.r()xTew,hxyoia)sumlsfidupecllledhe,antdhaattutorbaolltqyhueaxxdtareanntddicsyteahxreteenesleiomne.nts of F . Q 3 1 1 3 2 3 rest of this article, we will use finite extensionCs, that is extensions Rof finite degrRee. of .√IlfeEngntishooatcfnoaQrlelrx.ertsaIepftnoiEosninodiansilnsaognfwtFieotxh,ettaeohncnehseiqoeunfxeCitostrtsfaainFoceno,wloetwermhdeeceionaqnratuesti/e√tni3rssn2ut.atia.coenTntrnoIdehnatswhel/tloehrrSpdeda,noreiwiselrnueu“tistnstHoectaiootepolnhwrlrneeettsmshtceeoeurdrealovclQraehidst-ttbip h“nenumbersilHergagtcoaseewtrr(a(ufitixmscehetluesEdu care)trhoteaehtbxo iaiofotfbgnentgan aFeinfiniteslifirin?3oepi”nlstldahWEonewfe hdtdegreee)tha?ehntaseeinWdrefeodFedr w?mthoverpo”eaoevmriWne. athdtheF.kaioestrcs uderealsusxorsaeewmde p–al eweseg mcanisena-tn oefxtaennysiroantiofnal and so is Q( 2) is the set AoftatlhlrinasutpimoonbianeSltrs,ismawonifeldathrhilaesyv,foeoafrpddmrjeogaivnri+edineebgd2twaenoo+(iurnrcgoa2httitbohanewcakl2hgceriurnoebu√ean,d2rb)o,o.ocntAafiwrneolerduxaltdteihonlenesoairodlynsF.ttooiSefelapdvrsecogruvarbellieoac√wdte2wjxotoithsneeiinsnnsaogcictosatnlcmilo,eonadnoysroatbrofquenaucedatjidoobrfilanedtiiesncogglrevexeietnleg3mn.seiFnoontsr.. eIxtTahmeprlaet,ional numbers are a very convenient field for many mean by “Cbaoidgumdgliednrag”wn,?esbWucyboe“ntrbsrateigcrfutegircentrtg”op,?omtihWn3uetelstdireopegulfryeteisrenidtgoefatnEhdeodrvaeievgtorirfodeFneiQn,a.golofIfpatfEleraEennowedivs?enrlaTlon-Fdhte,axdfitotnf[ieteEsnednism:inotodFpneo]n, eosfindolfsw1eti/Fmbhe3dli, ecanht[u Ehtfinitesiesi2:no/qtFg3nuh]e ot,numbersnhdtwleiiyohmnitceehhwnwese iiofsfioatnetr lhaegeneratorsdoie,gifnhdhEttiemnreaedecssnegtasewidoa enaslioston a“above.fekH.Eeoawallsm aInu ctheh b igrestger is E than F ?” What do we performFediginuretsheqeu3De.nelcieat.nurFpnorsrooebQuxleta(mm√w.ep2)lceai,snlieftcnhwogenetshtetartmkuoecaftyQaQlblw(en√euc2om)nabsayetsrfiausrlcolstftnetduehn.melIabfreowgresmeothofaat+Qvhee(b√2tfo2r)cmo+annascd+t2rtub3h√cetnw2ahafupseroateihnaaetn,rdbint,obctvhQaaer(re√yrra2aot,vtio√ieonrn3at)alh.lpse.lraSanteeiv,oenwraellsma. daSjyiomisnitliaanrrgtlsyw,mitahysbeegments of vector HspoaAwcpteFovtovisehgvesriiceu,sbtrrewlopeFreoci.shn3opWtam.aC,vcbCeewoeiownenonasoivhslttletairornvuuneoFsccsettpt.diwitrnWroitgytvhheieatdowtecshsidoeelgmlesgemesnmnpcoeoaretnpiusbntsetgetrhosyatofothbtfisfioamqassncurqdekis.anaeugirsInrnafetcoucrrburileisluyobndorbedt“otiunmbtotlhgientegiensgrllfitgaeioetnbrnhli”ofdg.fo?rnutrathalmWhtll.ebaneooluurlrnytmefeabtelnosraesrbtepsosxraroaratanevmhdtadeedps√oilodnenne2.gsg..TarinesTnhHenwhieonoixwskftpaefiEomcoveiofenplnodrtlv,shente.eirtrauoTctiFcoushatmr,iianabcoplkhlvfleltyaeoerxsenftoyx.tldtvhrTeeiennshtdgcoirostimectdthpive[lEexo:nFe]t,owoh. icFhorisexthaemdpilme,etnhseiosnquofarEe raosoat of 2 has famously been proved Suppose F is a fiIefldF, Eiafdaajofifiinenliidnpt,egebreafyxontrtyemhneiesrldirFeoanin-tngpiotlsahnfenaqicteluoaserwqrneudcseampL.roeFnaordnorfiionFetgix2ta(,etnmoteohxpetaatltetc,hnisiasfiotaowflnoleiftpostaofaofcikEnpoe toe.thissQrTfd(exihwcn ,etarticle,eaynts)meq[sLuauyac:rnhfiFe drwet]sehtt=.hga ee.twill[nnL4bla)uo:r tswgEuseheeo]tx[utEh olafiniteedQn:cldFoe(√ao]y. dr2extensions,daT)tirnoheaainesatdleeqmtauhxeaeendnstr sfathattuotorifctohFb eeistx.ra titextenenonstQiho(en√-.d2e,s√ire3d).point. plane.thWeeDcealilal nitparopbl1laenme.because it is two dixmensivoencatl√oarsspaascpeaocveewr hFe.reWbeywoinllendotitmotebrnyestiorornadtweisoecnrraieblf.erAthtdoijsotiihneinfurgella(albdudtinelga)b√or2atteooQn, awneeoxbatmaipnlea. nTehwinfikelodf dtheneoctoemdpQle(x√2). Please note the new field is mayraotcpicloaunnraelo.snWalynedunciasdlelorfittdhaeegprfoleaellnotewbioneg(cnaCcouoitrsucetludhimtewsi2setaticnwocnoesdt2r(i)mup. creAotnvnpsiidooeienxdateslinaotssueiortanseisionssdcopetfaiotdchne esgofwrrdeh aofiniteeetirotoewcncoba uydegree.lirsp)o.lcnaaenllede? dimTaehnqasutiaoidnsraiwmteipcroesxfsetirebnltesoiuotsnhi.negIrteoanlly the straight edge alone. demonsItfraFteisfCtahafineedwldeeg, rSbceoiyemntasihltcaretrusSFlcyuit,n-ppaplaladlonjpsopereioenwFdiinnietcigstsmaaobenalfientierhwlFrdea,ytQ,Eiout-nhppaaaloalfitncncieusimtb(eautelhleltexrpipotoreolaientntsteiwiosxonotn(aeuxnlol,dsfypii)olltaensnaued(nca)dh?dtojLtoWhainaneotcifintuhbngbaoi)vlit.lcehreedaxxxtitsitaoecennundsasisiloyoesndnaw,roaioetfrhseEelooge.nmneTeehoneeftxnstdoroe[faLfgareF:neley.Fm3]rea.=nttFi[oLo√nra:2elE.xa]I[nmEop:rleFd,e]r. tTohpisreserve the structure of a field we need to make sure line. And itnudrleinnesde.o, u[ACtnw:deRicn]ad=ne3e2cdow,n[shCterru:ecRt]hQ=e(g√2e2nw)ehraaesrteoarltlshnopeuflmgatenhbneeee.rrcsWaootmoefrptcshlaeoelxlffonitthrumeamcbpao1le+mar3nsbpe√alerb2xeea2cn1sau3aumnsaebdneidirt.sbiOsavrntaewrty1ohoeadvnioemdtrhetien.hrseOihornanantatidlho,eantsaohlatseh.seSpriamhcaeilnawdrl,hyet,hree by one dimension we refer to the real For aFolronaglotnimg teimscehsoclhaorlsar(hs1a)hvaeAlvedebndgebitenehnegnmaQawnwy(ai√anrbrte2ee)rocdosiofsefnmcttsthtoieornunssacetfpftroHoeaorodrtoiefne.wmmastIelefotlvenhfwenterutiet,owimdnohwoebenagedvlerirehndcseatovsocna.efocscnttnohrssnouetitscrnfttuoriaurcosmetncpditsor.aeantd+hTspie.chbot2eciTanrobe/thmleiw+pnreawectrsah2eswyems/eourrapewfntoaiyhmonr.enfiarmaIgenltuyaplir,tslefiaibspt,ngilcwmeu,aerewrexetsetorewmanftoseaiiroyomnnsaat(llasalry.dtSjaowesivisnteihsrnagsale)da.gddmijnoegnintnisneogwfs pmoainytbseto a chart. This real nuCmanbeawrdseFrjaoeciragionelnuionsreftugrmiuna3cbfin.tenyraiCstilerolarnpdarseteotigironrufntecasietnliofinsnvngqeiutrahaetershdeeQgrgmr-orepoealetnalstco(en–voeof(wrtlsihenqtteheuhce.aranrtAretenoniaorfdlotnsaoaifi–ntlnpdwlpdeelreneaafdgenctcfi,aehtn)[C.?instqoe:WutRnafieur]nemh=daebv2.aegerw.fidon4haifi)esdtgcredweuinosntsuehgerml,edasbtgauoleerbrnasstedoragrfaasmtcgtoateoibnrasoetgvrq,ooaeufmf.taotadIuhrnrlsetyatiactphsriolceaymatiebinpoxoglnteveaxeanl.nsndIiuonmndt.ibhveirdsinagrear1eawndelli-.deOfinnetdheoontheleermheanntds,otnhethe new field, hence we take all couldcocuolndsctornuscttr,uhcto,wheovweervenronb(o2ob)doyAldeydkndngkitenhwegwcahopnhroerowriewfnsotptretomoornspepeddrcriotniivngoenstecopqeemeorurtaiatenacyantihniconoeocfif.fficgaFuigutorlsurirenorcseTenoxsalohayrranemdurdinpeSnoaldoanetet,lcoreciutsrofttcnhwcaliseonoet.ndrnftousatltcklhrtoeeuiwbnQciltneui.wgbs eTelceithm.rhecaeTruyemchwofiesFieldsoratrsraestdntwiecnhneae slallowstace(ratpgahlrelxeofte votheocsirdaQtelol d(action√ofoibn2rtt)aeairn snofedtch tadjoiningteihoednnessfdiuroertdo helements.cpecrouitrno)t..Q(√ The2, √ ra3)-. rest ofletnhgistShaimrrteiasclatylerob,lyfew,techaoidwsnjiosaltilrnrtuiincselgtee,fiadnwn.ietIiefrwrweaixletltioheunasveasieloficntnuosi,btce3etoh.enraxsortteoeraitunslcswenitoxouantumeslpnd,bosteilhronesaanttadsirniosetfoteohfixfaentieincrtnaufiestbnidioictenegsaerdxleoetepfeg.lnfirapesnoeiieost,osneivw,bdeeloreergmtcrhooaeenmeye.rbseotianaflrasdtt–eiwogwnritesehewcs3aie.tnghnmFotoehtrnefitsensexdoafmpaepfirlaneti,tioennsuimncbluerdionfggreanteiroantaolrnsuams baberosvea.ndIn√t2h.e This field naturally extends the (3) 3AddCinoguladnwinetceorsnescttriuocnt ppooiinnttss oofutwsiodecitrhcelersa. tional plane? That is impossible using only the straight edge alone. a moaremeolraebeolraabtoerattoeotlo, oall,ecnaognctochnecpoetprrtehtsahptaotnwdwSoionuugpldlpdtobbsrereaeFacahkisotrfaaidtfidsieitltcidio(o,no1nE.r).dAiandfiadtneiinstegaenaxdntetinh1ste/ien3ornsuesocetfi2tio/thn3eapncodoionLrtdaoi3nf.fiatnTtwiethoeare3lxaeinetxSisetosetn.lnoausliotsobinaotnaondifnEitsh.oeTf dhdeesngir[eLde :tpwFooi]n=(tn. o[Lt t:hEe][2Ein: F√]2. )T. hAisn extension of degree two is called a quadratic extension. It FiAeltdsthaiQsll(op√woi2nFt)hti,ieslwdtaehscethaisoallevnotewopftrahaoldevljinodauicenmtdinobengnerooesufloegamfhdtebjhnoaeitcnsfko.ingrrmgTeoshueatenle+odmrfaboet2nhioitsnfis.ae+alrdTctn2ihtuchlemee,orbawreytrheisoetwnoraeairlplelar,nuoabusv,emevcefib√arnyer2iertsceirosaentnrxvieoteetnnanaicselvosineon.trnsSytsfier,ecuvtloecdhnrtaaivfbtloelrnaiesdimesejonxaltitvneifiyinnesgglidsonmfsoaroyfmbfiaenniyte degree. HFoowr adoloynogutsimhoewscsChooomCmlaeoprtushulidthniHangwvgoeiwstcebhiovmeneprsindn,toertawsuemswreciosbatehnrlpceaest?otviorioenafTntnthseohpsetoetufiuahntsfectsoetidroeifet(mtg2thtirw)ehseeneoepActrioladoiamscnndtsteieipsonbsdaniglisneacnsaolvtansopopslivpatnrfrenraetusodrecv.?rittcsehiIteToetascnuhbtissicalo.ehtltnuTiwiamtsphaioesiyemtnirntauepotopeowfofomsoeftnsrweiamebnmoltlameiusnltiluaeomytnsiopaiuyaknlnlsegtfisdpaegeoensaxuonetraplcoeydilunsredstcwihialnoleei.gnsent(araeradwijegoqphiuntoaientditgogs)ne.tso3awlaohnciech.hart. This How do you shorewassoonm28s.e Hthpoiwenregefvoaerissrmo,nieimsdt. piisHnoaoswseviqebeuvrleyenr?,rceiTts.thiFrsieocartfieavxecartoymnitrpeelisetsto,roipifc.owtsFivesoeitrbaFolekenixeeatlQdmoColumbiatsopwpoal.relol,oFmvtwoheareytsehusfix qecrScienceaushmatacrpaetenlieoslr,antorato ghoReviewtteef motasofqdteu2juQnorathi(nrn√asetisno2orgo)ufoaktoametnwlpoedoemeuftcoseh2alpnyenlhtnebsca.ofesauenTrfnsatthmrpeurorcorutvtasoeQtldyi√Qo(√nb(a√e2le)2nn,au√psmra3obl)vl.eenrdusmabreeras ovfetrhyecfoonrvmenai+enbt√fi2elads afoarnmdabnvyary over the rationals. Similarly, wchoiuleldtocownsrtarpuctth,ethiroewmDecievnaledniarsnbnaoeprbordomuobdnanlydeye.mkobnO.cyecnwuetrhhseooimnwellpyetlmoeuneptAndrriotetcavrktrethyihcsiseotrpoftoeao(lre3ilnaso)twfiai,Aoibgwndulgisedrsehichansiargaafivcnrueaelmnidpnnrsivonotaatavtlrleniciodracowsennedstcst(oeripanrourdonciodnutvpidgbtiodhilcoienanb.dt,asoTicnrsokh.utfgebrFtrteowsoreuarwocnctaectdixsiooraontcnmhnsl,eepdsfim.lcoeealu,dolllwcttifcepohurlerioc)a.rytioton parnodvediv2isiiosnn)ot constructible solving while to wrap theitrombeiniHdrrsoawtaieSrovounetuparo,nlp.bwodAse.eedihOrFjroaanivintesiionansngiofiamt(lea.pludAls,edediEdntjorgtaih)ncefik√innc2giiostmt(eoatpeodQxadtes,iesnw2ntgssae)oiboo√flnbair2srteoh.aatfiosIniaotQnnaais,ni.wntdeiHvwemLaooerfiwbaietaetlofiavdnnietfndriot,oeraenmirtneoaixetinsletwldeyanifiQcasveaise(losdtr√neoysd2rore.)afae.nddEsFodPtjtor.ileniericTdngateihsiQnnvxeege(annw√moa[nL2tpe)yeo.l:teitiFhnP,rotreo]lwsae.=ntaetiesFow[eoLnarnafi:olceeEthxsledqa][rumtEithsa.per:leTenF,hre]owti.shoTtefieh(slniqdsouitasrteharotootfoaf p2ehrfaesctfasmqouuasrleyeb.ge.en4)pwroovueld lead to a quadratic extension. araemporerseeenltaebdowraittehtaoIofvlweF,riaytihfccooaounmfitcepetlpladitkc,aitbnthygeadtshqoweuboajFuerc-ledptrlt–abohnroeaetasDfiwkgoeeulrtiramaennd.eyaipWttnrihoeFinbnw.lge,omfutah.lndactliieiksre.atlOol npaostishnoetcsioa(ttxhe,eyar)mhsauncdh,tsthihmaetplbaeorttohebrxjtewacntodwcoyitnhasrteruecletimonens tms aoyf Fin.volve not allmraaytioncncauolstr woanlil(tl1hyr)aoutAnioednedaerixlnstghrwaeaitnefholeilnlmootnweenrisCnteegxo√ctmtcr2iiaopr.ncueIutlnpeminomosginrtedanttnheotorcef√betitseon2w(t.iporerIrralnesoietsnvioeceoirtdsnivd.eoaeednlr.tihptnAoetd3eispnrj.totrsreTiunoscchef2itntiretouvgwnreS(eosatohodllifeodnuSaiesotnistmcfiigrociu)eniunlc√vadrto)ru2.lwlyrvte,eosaonQdftehjea,odewifinsteeionllodgumbtwtaiaaonekinneniroesarfueandrtewiotwonafismliemalcdkuebldteasnunroreoetooeutdswQlion(u√elad2r)l.eqaPudlaetatoisoenanscowutbehiitcchhexntewnsifioenld, oisr one of degree 3. For example, are presented with aCvaenrywecocdmoenpmsltoircnuasctteradtlleospbtojhienecttds e–ognraeItefifhgFaeucQtirsfe-pa.inlfiaWaceeledp(,rwtebhdoyeiuctlrtdhaetblilioFekn-ewaplltaaoypnleaunpswseoo)en?cmimaWeutaelntiahpFalmev,euetxdchthiaestcnsusisimosanepldl3l(epaardosijonoebgtinsjmei(ncexgnt,)ty.wo)iftshuacnhy trhaatitobnoatlh x and y are elements of F . it, foHr oexwamdoplyeoau nsuhmowbesro.mTehtheninwgeisstiumdpyotshsieblwe?ayTthies fnaucmt ibterisvpaorisessibalse wtoe epxrotevnedsuthche fiagsutraet.emIfewnetQctao√nokcopnecolpuldeea 1/3 2/3 it, for example a nuamddbinerg., sTu(hb1et)rnaaAdcwdtiednisgn(t,2gu)msdauuAnybltdtitrnidhpaticelnytrgiwsneagcny,taimoitnnnhdutielpstrdoiscinpeivanlucniytdmiiinobnbgfnegetaprwdanorovdienanlertwidnniioebevltfysil.d-aaditlnsehlifigenwrnaeaeetrladieonmxnodwtaneelanslnetlc-ldawedirrmieytctfihleenone.oeptfidnsegeroaunetrxioetteh.nrleaesImfenae(lweenfiwmete2selcfi)dnoaeitnsnlad√tlt,clhho2ohewe.nessncIenlcet(uewaooddfwrefiddaeeieltlrtdi3oan,tnkuohe,mepsabnruleclbressterwroavefcettihtaohekneef,osamrtmlrlulcatti+uprlbiec2aotfioa+nficae2lnddwdweihvnieesrieeodna)t,ob,mc akree rsautrieonals. Several adjoinings may be thwahtilseomtoewnruampbtehlreesnirgatrmheimnudnasaytatbraeoinucanobndls.etrOiunncetseuAsdcit.hmItCpafhlaiwefsnaetsprwhoicieaioknvncteio,stnwtotsheotecrnohuenascwtvstaterabuplwlicrsotophuvoalaiidnpnetoksdiinnoeovtnwaiorntiuthahtgenhahteQtbor-traaphctliekainogcdfinerigaocu(lautrtnpheodlesra.ncoraoenF,troirfiworeenselepadxmolatnpamhdyleaiponslrtgeya,)rt?wtooewWpietrhohvsaeevg√em2deinissctsunsostfedcoansstergumcteinbtleosfoalvniyngrational possible co(2m)bpiAonsdasdtiibion(ln3egs)cawoAnmitdibhnditintenhargestseiaoecnntoisoipnwnetriepatrhotswieiotnicthnthesiosiefnuntacoplptuolaiedinkrnieaitnantsgadigondrnfdsiansqtagwuiion,aocncrsialerucuclibdrlnetcoiur.nloeamgtscs.btriaoentrgiso,aannmnaydluthnl√tiuinp2mgl.ybfTieanrhngpsicesairefinfroed.rldm√Odnie2nvda.itdTiuhninrheasgiesollqatyfihureeelxrndwtchenenala.lndt-Fdusoer,trafihtlnehlyxeedaelmxaotpnetleneerd,lesitfmwtwhoeencttosankosentrQuthcwteieonnmeswamyfiafieylrdsit,nehvneolnlavcreegewteotQak(e√a2l)l and then further to Q(√2, √3). that tshaoermme epwrneosuuemlndtbenedortws iblateehrnegcatouhnvnescatrorytrutrcaeotisimnpbaolpebnlildeceiatnthigenedtrths.oouebTecDajhhececihslatlieaoi–nsfnfagapistpthrsfireoigcmobuionslaroe,eyrml.ydtb.hiWnetehaecnetoewnpwsosoaeutirnludwdtcolottieufkhdelvd.n3iteI.okuwfTnsaweswoheswteohcaeSitravhcoeteoaelgotuaortdtincmhiongeunanctstfihotegrstuauaixcmrkteeepssalitenpcorotorihnbretetjaesircpnientsottwnthhodeietfihrndagteisotirnoeadl ploainet., we may start with segments of rationCaolsulad(n3w)dreAaisCtcdioodnfnmisndatpglersuugatracnietnedgipntotwisiehnoreotslsf(eincndoottgueiteogttrhsnrtsiehedcepceeototr2iitwornheniotensspp(r√onoafnoit2ntid)owt.itnohaAgfelpcnttip2oorwlescaisoxelneniatbslec√i.l?nhe2secTois)of.hnmiinaAtbvotsnoifinclsdevoa2xeiotmgstiroerdptneihoensesiawotstwsineibotoslhoeuafitsunidhosdceinangstlSerghloeuoefdeopnptnptwealwuoryosaqsoeutsihiaitoFsmedhncreusiastlcitlrtnliaoecconidfilgreuhedxaodltitudineqnes,naudgEtsalgieirdnoeaaraenataxi.fitloreioncnIsneitatqteelo.uxnteaoeuxtbnmitotseanbinoisesnnriwo.stnhaIetinocdfhdie√tsia2rn.edTLphoisainfifite.nlditenaetxutreanlsliyonexotfenEd.s Tthheen [L : F ] = [L : E][E : F ]. This themthiwte,oafuorrltdiecxlnea.omtpbleeacnounmstbreurc.tiTbhlenewitehestru. dTyIhfthiFse iwisfaapyrfiteehclidiss,enbluymttbhheerFvpa-oprilinaesnteaoswfweveimeewexatenwnFed at,hrtehfiaggtouinsrega.lltIfopwotieanktcsaen(ixnc,oytn)h3cseludrchesthoatf both x and y are elements of F . turns out we ctuaAcnrtancnstohonibussetprwduooiecnntcteQa, nw(b√yceo2hnt)hasavetrseueapclletrlmonQveu(inmd√teab2dre)yreasnsoopafuelrtglrahhnatetuiboifmooannrcbamskelsgraasrao+onfiufdebntl√dhise2oaofnalfolsorfidmwaelsgadarn(e+dtaehdbbet√dwovir2taoyiroa(yntsndoo,oaevmtpsaeurtrnoobhndtvtehserbtea2r√vcaraitatn2iretoyisi√noso,tn2vhnam)eol.rstud.AtlcehtSongieipnrmelrsixaeitclrtaauietrocncilnotytss,anibolislnae.noaSsdfoipmldvdreieilngvadrgirisecliyeto,antbw)leo wisaycaullpedonamqualtdipraleticexetxetnesniosinon(a. dIjtoining). the artthicalte.some numbHeroswCaevroemru,pnwuatetitnhagainvtaehbenleointCitnuearsnsseuedwchtetihocaenoCfncpaososotmuhirnliupdotcantwso,sfaetslthlcoweoponfnoaslirwtin.nretuesIsctwotionnspuvotlodihimnlevktesnQsto-otwuphtlfeasotichrdsemaoetl(autlthtlhiyeoenarrfiasaostgtefiiuoostrnnweaaasdolldcppsoiilnlmaragrnneuenesl?tp)ea?woTnnWehpdoaioeuntisnghitslatsivniomteeoapdraoissecsqhuiubasalsreettid.uosnTaisnhsgwiesghomniclehynthoef satnrayigrahtioendagle alone. admjoaiyncionacgncuabtrnheyeoadndiDowrljyorneiaitlenuithainobnduyngeptartarlthontasebhyqkleuieinmarlfgeorrma.leHlstoqeoirwonuowntioaenatrvgrleey(srcnrq,ioruwoptcatuetrsrmhehaoastartivooaaeonntfsncyea(otasnthupo(ifiunrtepesnrgrletfsodehdfvocaitauntdhltcseloeoiqdwefwucreia.osncrmtpa(Oeeanpernedrsac.fedsgetocsi.hcntteis4soot)snorqfut,swauhcrodsaet.uorQeblIdhot(erca√.ilacngseuc2ad.tr)d,i4)oma.)tntsheow,eatolamlouanuqltfdulouttemrailrmpdeblatraiewcadlraltsoyitcoicoafoensatxnshetsaqestnunrfadauosddicrodmtrniaionv.tangisc+sinoembenx√wa)tye2npaisonsiiovnanotals.vntedo bavcahrayrot.veTrhthise rationals. Similarly, them would not be constructible eithleern.gtThhmisaiys2pb. reAecloisgneeslytbrrauthcteedp.oiInftwo2ef hvaievwe twoecoanrestgruocintgatpootinatkeininthteherarteiostnaolf plane, we may start with segments of Similawrliyt,haodujtSoIitifmnaFikinliagnirfglayans,qfiaiuerdlardjaro,etibnioryionmoatgahtlsyaecnouForbci-rceaprunlararytonitooehntnilwnaywgleoucumfnualddebneaecarlnieedrtarFojhd.ooeitnO,tfoiwotnhloaglaotuthwcaldeuinisnbyolgaietcilahrlcderipraxrcottuohienomanntansssaidtcol(a,unxsnbt,,qchiyuocee)raserslo(xaeupntcrtehoenrvostitfhtdowadn(etndoe,goboicrtonroettenhoehsrn3atxse.rteucaoFctnftoidoirdaonyenespgxsardearemromefeaoepc3ycltl.eecim,usnFqrveo)uonr.altvrseexoaefm.Fgp..l4e), wou3ld. Tlehade tSoolauqtuiaodnratic extension. the article. 3 (1) Adding3 an intleernsgetchtiocnorpreosipnotnodfitnwgotolinee1asc.3h of i2ts3coordinates and then use the coordinate axes to obtain the desired point. ´ Q √ CaQn w√e construct all po2i.ntAs lognetbheraQ-p/laScime i(ltah/rely,raa1t/di3ojonianlinp2gl/a3anne)?irrWateionhavl ecudbisecurossoetdwaouseldgmleeandt toof ancyurbaictioenxatelnsion, or one of degree 3. For example, Evariste Galoi(s (21)81is1-t1h8e3s2e()two2fa)sailasl ntbhuremillsCbieaeotnruotslfdyoaofwlutlhennegucofmmonrbsamettrhusaecmo+tfabptt2hoiceiniafton+srmoacu2ntads+iadwbefi2htehercree+rapac,to2bilo,itnciacaawlrlpehpleraranoetepiao?o,nnbTae,lhncsa.ta.tSreTisvoreiadmrtaaipyolonahsadseljisob.ilneSienuvgseisnramglaoayndljbyoeitnhinegsstrmaiagyhbt eedge alone. 3 (2) Adding an intersection p(o1i)ntAodfdainlgineananindtearsc3eicrctiloe.n point of two lines. At this point, 1w/3e hav2e/3provided enough background on field theory to prove √2 is not constructible solving ´is best known foprerhfiosrmwoerdklienonnpsgetGrqhfuaomleronmiacsyeedg.brFieonoucrpsoesenqxsiuntaermunthcptele.td,Fh.ifoeIorwfrewyxteaoamfhkafiepvelQeld,tswio.fQewcW(moe√nhats2iyalt)ekrfiiuesrctsQtuthdaewynpesilneaomtgringoafitefyeitalnfiodllrtQsenhtxu(ete√menrna2blast)eirroagsneoasdt,flothtphhQeleae(nwn√feoaf,u2rsm)rwataelhsaneom+drattbyoh2esQnta(f√+rut2rcwt,2h√iteh3r)tws.oehgQemr(e√na2t,s,b√,ocf3)a.re rationals. Several adjoinings may be Evariste Galois (1811(3-1) 8A3d2d)iwngasana ibntrHeilrolsiweacentvtieoryn,owpu(oe2n2i)ng.htaAsmvdloeadgftniethnowbegtoramuacsnaiertdicincletieahsr.nesecacontmidopnapsfisoeisnrotcfeoafrp.aoIlltitniescaatlinmdperaottpochiorfeocnlDreemn.etlai.lalnyTopasdrsoaebysleahmded. ing new points to a chart. This able to come toSaupsipmopseleFsloeilsnuSgatuitopfihnpeclodosf,rerEteFhsepaiosDfinanedliifiitaneengledaxt,ontEedenaasocitohfihnneoiroftfesitiiemstxcaitloneaondrrspdLiporinoearbaoffitlofenrmsimittaseea.ndnedBxdientLfehosnereasneqiofiuuwnneseneiotcfeteahE.enexF.dtcoTeoernsohcsereirxdoniabinnm[eLaoptfeh:lEeeFa,.xh]ifeTe=wsahre[teLtontoa:o[fkbLEet]a:Q[EiFnw]:te=hFem][.LdaTey:shEfiirirse]s[dEt pe:noFlian]rt.g. eTthoisQ(2√2) and then further to Q(√2, √3). is best known for hisCwomorpkutoingGthaeloiinstegrrsomeucatpiysonoincpcoutihrne(ot3nt)olhfyAetouwdnrdoydineloirgnfetafihsneinlifdnvotsolel.lorvwsWeesicnhtgihiolecnirsspcotouluuimndttiysostinanongfocftefiwstewo(ldpocrisreoicxvmliteduese.lntdsainioneIntofesurF,sehilcfeitnaiewoafinaressledaq,olusbaooytcictouhnres).Fw-hpilcahne we mean F , that is all points (x, y) such that both x and y are elements of F . his mE´evtahroisdt,ewGeadnleoemiesdo(n1tso8t1rian1t-1erso8dd3tCe2uhmo)ceueowdlndwaesgsthwraaetetbceaarosicnfilttlsehistalerdniundticseaty.goprueorneeingdaticmscttosautbihntlseiamdwepaartyteihcdueSiiapcurnotpaantpabinomledsneuawallFtaipfiypilelsareuncapeox?fipnteeoTlmdnlhi,stuaiiEoclttaniilaps(lpafieimrdnoejipxpotoetionesnensienxisbngtiolte)e.n. suT(isoaindjgaoyfionnihitlneyagn)t.dheLstarafiingihtte eedxgtenaslioonne.of E. Then [L : F ] = [L : E][E : F ]. This able to come to a simcapnlebseodluotnieonbyofththeeelDemel(ei1an)ntCaAoraydmnddpoinupogtetirnhagnetirotinhsteisemriansiletafiecretsilodpencrtaoiplobloonliwenpmtsoos(in.fatdtBwdoeioftfiotlowirneo,eswls.iunebecstariancvtdoioelvnsCec,sarmintbhueweltetisphoclleoiucnthasiteotiranourncottfaotanfwldlopdsoiivimnistuisoltnoa)nnetohuesQli-npelaarceeq(uthaetiorantsiownhailchplane)? We have discussed a segment of any rational is best known for his worHkoowneGvearl,owiseghroauvpesniont tuhseedthtehoerycoomf pfiaelsdssds.eomWfoahnri.sletIrtsattiusedstyitmihnegdtfioeeglfdroeremxataceltlnyssiaonsnsase,sphraeeddwdiciantsagabnllseowwapyoiunptsontomaulcthipalret.exTtheinssion (adjoining). his method, we needwtiothionuttrotadkuicneg wsqhuaatrearfiooe(2tlds)coiAasrn.dadbniyentghdaionngeinfbatynercstiehrce.tieoOlnenmptoehninettaorotyfheaorplihenraeantadion,ndtshaeacliafirtectledr. altelwlnoogwtcshon(masadtryduibctiteoioncno, snssmutbrautyrcaticentdvi.onlIvf,ewmeuhltaivpelictaotcioonnsatrnudctdaivpisoioinnt) in the rational plane, we may start with segments of able to come to a simple msoalyutoiocncuorfotnhleyDuenldiaenr tahnedfotllhoewrinsigmciilraTcruhpmerostSbalonelmcuesst. i(BoTpnrehofoevriedSeowdleuinctatienorsndeecstciorinbsedtoheochceuarr)t. of (3)wAitdhdoiuntg taankiinngtesr3qs.eucatrieonropootisnto3sr. oafntywthoincigrcflaens.cier. Onletnhgethotchoerrehspanondd, itnhge tloatetaecrhtwofoitcsoncostorrudcitniaotnessmanady tinhveonlvuese the coordinate axes to obtain the desired point. his method, we need to intro(d1u)ceAwddhiantgaafineilndtiesr.section point of two lines. The Solution 3 Could w3e co3n.struct points outside the rational plane? That is impossible using only the straight edge alone. At this poin(t2A,)wtAetdhhdisaiCvnpegoompianrnptou,vitnwiidtneeegrdhstaeehcvneteoioiupnngrtohepvrobsidieanecctdtkioogefnrnoaopuulnoignidnhetobanoanfcfidktewgaldrocotiulrhinceldeoes.royinntvfiooelplvdreostvhteeho√ery2soitlsuotnpiortnovcoeofn√tswt2rouiscstinmiboulteltcsaoonnlevsotinruugscltiinbelearsoelqvuinatgions which However, we have not used the compass so far. It3 is time to formally asses adding new points to a chart. This the Delian prtohb(e3le)DmeA.lidadnicnpagrnoabbnleeimndt.oenrseecbtyionthpeoeinletms oenf ttawroyActoirptcheliresast.pioinnst,awfieehldavaellporwosvi(daedddeitnionu,ghsubbatcrkagcrtoiounn,dmounltfiipelldicathtieoonryantod pdriovviseio√n2) is not constructible solving If F if a field,IfbFy tihf eawFfiite-hplodlau,ntbeytawtkheienmgFes-aqpnulaaFnre2e,wrtoheoamttsteihosaernalDaFlnep2ylo,itaihtnhitnaspgtr(oixfsab,nalyecl)mliesp.ruo.cihnOttsnh(atxht,ebyo)tmshuaecxryhahotnahcdncaudtyr,baotorhnteehlyelxlauetmnatdneerednrttyswthaoerfecFfoen.lllesomtwreuincgtsioconirfscFumm.asytainvceoslv(eprovided intersections do occur). Computing the intersection point of two lines involves the solution of two simultane2ous linear equations which Can we constCruacntwaell cpoonisnttrsuoctn atlhlepQoi-nptlsacoen (tthhee Qra-tpiloanIcfaelF(pthliafenaerfia)?etilodWn, aeblyhpatlhaveneFd)i?-spclWuasneseehdwaave(e1ms)edegiAasmcnduedFnssitne,dogftahanansteyingistrmeaaretlsinloetpcntoaiiflonnatsnpy(oxir,naytt)ioosfnutacwlhothliantesb.oth x and y are elements of F . can be done by the elementary operations a field allows (addition, subtraction, multiplication and division) length may bleencgotnhstmruacytebde. cIof nwsetrhuacvtedt.oIcfownesthruavcteCatoapncooiwnetsticrnounctshttearurpacotionanltlaipnl optilhnatenser,oantwioetnhmaela(Q2py)l-asptnAlaaedrc,tdewwin(etigthmheaansryeaigsntmtitoaeenrrntastelwscptiotilfoahnseep)g?ominWetneotsfhoafvleindeisacnudssaedciraclsee.gment of any rational without taking square roots or anything fancier. On the other hand, the latter two constructions may involve length correspleonngdthingcotroreesapcohndofinigtstcooeoarcdhinoafteitssacnodolertdnhigenntahtuemsseaaytnhdbeetchcooeonrdsutirnsueacttteheeadx.ceoIsfotrwodeionhbaa(tv3aee)inatxAotehdscedotinodnsegtosrbiaurtnecadtiinapttoephironesitend.ctteisionirnetdpheopirnoatintsitoo.nf atlwpolacnirec,lewse. may start with segments of Could we conCstoruulcdt wpeoicnotnssoturutscitdpeotihnetsraotuiotsniadlepltelhanengert?ahtiTcoohnrartelspipsloainmdep?inogTsshtiboalteeaiuscshiimnogpfCooitsonsmliybcpolteuohtrueidnssiigntragtathieogesnhliatynntetdedhrgtesehesceattnrlioaouningseeh.ptothienedtgceoofoartlowdnionea.litneeasxiensvotolvoesbttahien stohleutdioesniroefdtpwooinsti.multaneous linear equations which However, we Hhaovweevneort, uwseedhatvhee ncotmupsaesds tshoefacro.mIptaisstsiCmooefualtrdo. wfIoterimcsoatnilmlsyterautscostefspooarimdndtasilnloygcuantsnesiwdebseepatohdiednotinrnsaegtibonoyenawatclhhppeaolraietnnl.etems?TtehTonisthaaarctyhiasorpitm.erpTaothsioissinbsleaufisienlgd oanllloywtshe(asdtrdaiitgiohnt,edsugbetarlaocntieo.n, multiplication and division) may occur onmlyayunodcecrurthoenflyolluonwdienrgtchiercfuomlloswtaincgecsir(Hpcuorowmvesivtdaeernd,cwienset(ehprasrevocvetiindooendts uidnsoteedorcstechcuetriw)co.ointmshopduaotsostcascokuirfna)g.r.sqItuaisretirmooettsoofroramnyatlhlyinagssfeasncaidedr.inOg nnetwhepootinhtesr thoanadc,htahret.laTthteisr two constructions may involve (1) Adding a(n1)inAtedrsdeicntgioannpionitnetrsoefcttiwono lpinoeinst. omf tawyoolcicnuers.only under the following circumstances (provided intersections do occur). (2) Adding a(n2)inAtedrsdeicntgioannpionitnetrsoefctaiolninpeoaindt oaf cairl(ci1nle)e. Aandddiangciarcnlei.ntersection point of two lines. (3) Adding a(n3)inAtedrsdeicntgioannpionitnetrseocftitownopcoirinctlesso. f tw(2o)cAircdldeisn.g an intersection point of a line and a circle. Computing thCeominpteurtsiencgtitohnepinoitnetrsoecf ttiwono plionienstionfvotwlvoesl(itn3he)esAsiondlvduoitlnivogensaonthf ietnwtseorlsuseitmciotuinoltnoafnpteowoinuotssslioimnfeutalwtraoencqeiuoracultseiosl.innseawrheicqhuations which can be done cbayn tbhee deolenme ebnytatrhyeoeplermaetinotnasrya ofipeelrdaCtaiolomlnowspusat(ifinaegdlddtihtaeiollnion,wtesrus(beatcdrtadiocitniiopno,,insmtuuboltftritapwclitocioalntini,oensmiuanlnvtdioplvldiecivsaittsihiooenns)aonludtidoinviosfiotnw)o simultaneous linear equations which without takinwgitshqouuatretarkoiontgs soqruaanryetrhoiontgs foarncaineyr.thOicnnagntfhabenecoidethro.enreOhbnayntdht,heetohteehleelmartehtneartnadtrw,yotohcpeoenlraasttttrieournctsiwoaonsficeomlndasytarluilnocvwtoioslvn(esadmdaiytioinv, osluvbetraction, multiplication and division) without taking square roots or anything fancier. On the other hand, the latter two constructions may involve WHY CAN’T WE . . . ? 3

Think of one’s mathematical education. For the most of it, we study certain sets of numbers and operations on them. By the fifth or sixth grade, we learn all about the rational numbers Q. A bit later, we encounter π and e which do not fit the old scheme. They are both irrational and form part of the biggeWr HseYt CoAf Nre’TalWnEum. .b. ?ers R. 3 Later, we come to realize one cannot compute the square root of -1. Then, we are forced to extend our horizon to the set of complex numbers C. Think of one’s mathematical education. For the most of it, we study certain sets of numbers and operations In reality, all three sets Q, R, C look oanlikteh.emT.hBeyy ctohnetfiafitnh0oransidxt1h. gAradddei,twioen,lenaerngaatliloanb, omuutlthipelircaattiioonnalanudmbers Q. A bit later, we encounter π and division by non-zero values are all well-defienwedh.icFhudrtohenromt ofirte,thmeuoltlidplsichateimone.isTdhisetyriabruetibvoetohvierrratdidointaiolna.nDd efsoprimtepart of the bigger set of real numbers R. the fact all three sets vary enormously inLtahteirr, wsizeec,otmheeytoberaeraltizhee osname ceasntnriocttucroem–puwtheatthwe esqcuaallrearfioeoltd.ofT-h1.esTe hen, we are forced to extend our horizon objects together with a few more form thetofrtohnetsleinteoWfoHfcoY(amCbpAsltNerx’aTcntWu)mEalbg. .ee.rb?sraC..They are so important nowadays one 3 can hardly imagine any branch of mathematIicns rweaitlhitoyu, taltlhethmr.ee sets Q, R, C look alWikHe.Y TCAheNy’TcWonEta.i.n. ? 0 and 1. Addition, negation, multiplication 3and There exists a notion called field extension. We know that Q R C. In other words, the rational numbers Think of one’s mathematical edudcivaitsioionn. bFyornothne-zmerosvtaolu⊂f eits,aw⊂reeasltluwdeyllc-deretfianineds.etFsuortfhneurmobreer,smanudltioppliecraattiioonnsis distributive over addition. Despite are coonntatihnemd .inBtyhethreeafilftnhumorbseirxst,hwghriacdhetTi,nhwhetienufaklrecnatorafanrolelanclteloh’sanrbetmeaoiaunstetehtdteshmivenarrtayihcteiaeolcnnoeoamdrlmunpcoulaeumtxsibolnyneur.imsnFQbotehr.restAi.hreIbsfimtzoenol,asettefihore,flydwitbe,iseweacnerocntoshtuuaenidntsyeeardmcπeeratsantirdnicsteutrseo–f wnuhmatbewres caanldl aopfieerladt.ioTnhsese C R R in anoethwehr,icwhedcoanllotthfietlatthteerolad(sficehledm)oen.xttoTehbnhejsemeiyco.tnasBroteyof gbtehotethhfieforfitrwhrmaioetirho.snaiFaxfolterhawnegxdmraofmdorerep,mlfweopremalreitatshroanefnfatrehlolxenattebnilogisungioteenrtohfsoeef(tarabotsfaitornneadacltls)onauilmsgebberras.QRT.hAeybaitrelastoeri,mwpeoretnacnotunotewraπdaaynsdone Q of . LIfaEteri,swaen ceoxmteenstioonreoaflizFe, otnhe cqauenenswtocihtoainchowhmdaeporadunrlteyoetiitnmhfitetargsteiqhsnuteeeaodarlendiysrso“bcoHhrtaeonmowcfehm-.1ou.TfcThhmehyabetinagh,rgeewmerbeaiotsaitrchEes ifwtrohriratchtneiodoFunta?o”ltheaxWenmtdeh.nafdotrdomuorpwaherotriozfonthe bigger set of real numbers R. degree WHY CAN’T WE . . . ? 3 mean tboyt“hbeigsegtero”f?coWmeprlexfernutomtbheers CL.atero,fTwEheeocrvoeemerxeFist,tosorafetaennloiztdeioeonnocetaecldlaen[dEnfio:etlFdco],emxwtpehunitcsehiotinhs.etWhsqeeudkairnmeoewrnostoihotanotfoQf-1E. TaRshean,Cw. eInaroethfoercwedortdos,exthteenrdatoiuonr ahlonruizmonbers ⊂ ⊂ vector spaInceroevaleirtyF, .alWl tehwreiell sneotts tQry, Rtto,oCdtaehlrsoeecorsckiebotnaeltotiafkhiecnis.oemdiTnpihnflueytlxlhnbceournmtetaeabllieanrbus0omCrab.nteedrso,1nw. ahAnicdehdxiantmiotpnul,ren.neaTgrhaeitncioknot,afmitnhueeldticpionlmictpahtleeioxcnomapnldex numbers. If one field is contained plane.diWviesiocnalbl yitnaonp-lTzaehnrieonbkvaeoclufaeuosnseaer’isetmaislal twwheiIeonlnml-dadrainemtoafiicletniahntelyesd,ireo.,danFWwulaluceHlartYathchiasroCelnalreA.mtNsshFp’oeeTaotrsecrlWa,etQtmEhwt,eu.hRr.lmet.,ar?iCpoe(slfiibtlcoeyaooltdfkoi)ointnae,lxiwdksteidmn.sisetitTunordshniibyeouoyncfteicwtvrohetenaeotrivnafeoeifnresrmerat0tedsordao.nitftFhdinoeoun1rmr.ebDaxAleaedrmssdppiaitlnieod3nCo, pinseeraganattieoioxnntse,nmsiuolntiopflicRataionnd asondis R line. And indeed, o[Cn t:hRe]m=. 2Bywthheerefitfthhe ogrensiexrtahtogrrsaodfe,thweecloemarpnleaxll naubmoubtertshearreat1ioannadl ni.umObnetrhseQo.thAerbihtalnadte,rt,hwee encounter π and the fact all three sets vary enormdiovuiosfiloyQni.nbIyfthnEeoinris-szaieznreo,evxtahteleunyessiboaenraerofatlhFWlew,HsteYahllmeC-dAeqeNufis’etnTsretiWdciot.EunFr.wue. r.e–?thawererhmaintotrweere,emsctaueldlltiaipslfii“ceHaldto.iwonTmhisuesdcehistbriigbguetrivies oEvetrhadnd3Fit?io”n.WDheastpditoe we real numbers are oef winhfiinchitedodengorteefiotvtehtrhetehoelfdarcestcahalsellm–tehw.reeTechasneeyntsoatvraefirnbydoetanhofiirnrmritaoetuinosunlymalibnaentrdhoefiorgresmnizeepr,atrhoterosyfabtsheaebrbotivghege.esrIansmetteheosftrriecatul rneu–mbwehrastRw.e call a field. These objeTcthsitnokgeotfhoenrew’sitmhaathfewmamtiocraeWl feHodmrYumceCaAttnhNioeb’nTyf.rWo“FnbEotirg.lg.itn.he?ere”om?f oW(astbesortferfaietc,rtw)toealtsghteuebdrdyae.gcrTeerehtaeoyifnaEsreeotsvoeorfimFnpu, moorfbttaenrst3danenondwoatoedpdaeyr[Eastoi:onFnes], which is the dimension of E as a rest of this article,Lwateerw,iwlleucseomfineitoe erexatelinzseioonnse, ctahnantoist ecxomtepnusitoentshoef sfiqnuiateredreogorteeo.f -1. Then, we are forced to extend our horizon coan thhaermdl.yBimy atghienfiefathnTyohrbinrskaixnotchfhogonorfaeb’mdjsveeaecm,tcwhstaoettrhmolegsamapetrtaianhcctseiarcwloalwvilateiehbtrdhooFuuact.tafttWethiwoeenmmrw.a.oiFtlrilooernnfoatothlrmetnrumytmhotbesotefdrroseofsnQcitr.,ilbiAwneebtsohitfiusl(adaitynbesrcft,uerwrlaltceabtieu)nntacsleeogltuaesbnortoaefrr.antπTuemhaoenbnyderaasnreaensxdoamiompppelero.arttTiaohnnitsnnkoowfatdhaeycsoomneplex Fields allow thteo athcteiosnetooffacdojmoipnlienxcganneulhmemabreednrlsytsC.im.Taghieneraatnioynbarlannucmh boefrms atrheeamavteircys wcointhvoeuntietnhtefime.ld for many Think of onee’wsTmhhieacrhtehdeomxianstositcafilntoeontdthuitoehcnaeomtlcidao.lnslBec. hdyFefiotmhreeltd.hpfieTlfatxhmnteeony.srstaiWsoroinexef.tbichtWoa,gtlwlehraiektidrnsearto,auwptwdiloeyathnlceaeeatlrbratQneancidanalulfsRosaeerbtmsiotuoCpiftsa.ntrthIutwnemoorfbatdethtihrimesoerneabwannilsogdinrgodouenpsmra,elsbtreahetsrtesiooarfQansrts.peioaAnlcaenbluiwtnmhulabemtreeebrrse,brRwys .eonenecdoiumnetenrsiπonanwde refer to the real reasons. However, it Iisn arevaelrityy,reasltlrtichtrievee TsoenhtesrtQeoe,oxR.i,sFCtosraloenoxkoatmiaoplnilkeec,.atlhlTeedhsefiqy⊂eulcadoren⊂xtratoeinots0iofnan.2dWha1es. kfnAamodwdoiuttshiolaynt,bQneeengaRptrionv,eCdm. uInltioptlhiceartwioonrdasn,dthe rational numbers on them. By tahLreeafitcefotrnh,twaoirensecidoxmtihnegttrhoeaedrwreeah,aliwilczheneuldoemoanrbenecoratsln,lfinwatloibhntoiehccu.ehotAmoitnlnhpddetuustricrenahndtteiehamoerendeas.,clq[oTnuCnuaht:mreaeyRibnr]eaeo=rrdoset2iQbnoow.ftthA-h1e.rbicreiTortatmhlhtaeipeotnleng,erxaew,nlwneeaurenaaWmrdetenHbofcrYefoosrrrsuCmoc.nAfeItNdptfeh’arToteronπtWceeoaExfifnmted.tepl.hdn.le?dixsboincugourgnmehtrbaoiserneirzetsodoanfrere1aal nndumi.bOerns tRh.e other hand, th3e to be irrational. Addijvoiisnioingb(yandodnin-zge)r√oav2raetlouceoQsn,atawreienaeoldlbwtinaeilntl-hdaeenrfienaweldnfi.ueFlmdubrdteehrnse,ormtwehodricQeh,(m√inu2lt)tu.iprPnlilcaeaartesieocnonnoistaediitnshetedrinbineu⊂wttihvfiee⊂ocldovmeirspaldexdintiuomn.beDress.pIifteone field is contained e which do noitntofiattnhtohetehseortl,dowfsecchocemamlplLelta.ehtxTernhl,uaewtmyteebarceroreasmb(Cefio.tethlorde)irarerleaaxnlttiuizeomennsboaineolrneasncoadafrnetfnohoorefmtifncoporfimanriepttreuo.tfdeFetotghhrreeeebxsiaoqgvmugeaeprrletesherCeotoroitesfaoralfsena-–1le.xwntuTeemnhcsaebineo,nrnsowotReffi.aRnrdeaafnodfircnseoidteitsnoRuemxtbeenrdoof ugrenheorraitzoorns as above. In the not all rationals wtihthe foancet eaxlltrtahreelemseetnisntva√anr2oy.theInero,ormrwdeoeurcsatlloyl tpinhretshleaertivrteesrtihzae,(sfittherleudyc)tbueexraetreontfshiaoenfiseaolmdf etwhsetrnfiocertemudreetro.–mFwoahrkaetxsawumerepclaellCa ifisealdn. eTxtheensseion of R and so is R Later, we comoeftQoI.rneIfarleEiazeliistoyan,nealeclxatntohennroteshteioecnsoseemottsfpoFQuft,,ceRtohtm,heCeprqelslesuoTqtxeouhsoknatifiunroatemknhlirbiokwosefoera.otsrnatToCreifceh’.sl-ei1nmy,.twaecTrteoehnwsettmnaieli,dlanwutiis0ecea“alfiHrnWenoddHiwftuYo1ecr.maceCextuAidtcoNedhn’tndTo.sbiitWFieoigoxngEntrsee,.rtn.thn.dihse?eagoEmtautiortsisohhtenaoxo,nrtfimezFinotu?s,nli”twoinpeWslsihcotaufattdfiiyondniocteawrndtedaeignresee.ts of numbers 3and operations adding, subtractinogb,jmecutslttipolgyeitnhgeranwdithodfiavQifd.eiwnIfgmEaoriresewafonerllem-xdtteehfinenseifodrnonoontfleFinle,emtohefen(tqasbuosetnsrtaitochnte)wnaeelgwaerbfiereailn.dt,Terheesnytecaderewises“oHtaimokwepoamrltluacnhtbniogwgeardiasyEs otnhean F ?” What do we to the set of cmodmeivapinlseixboynnub“ybminbgogenresr-”zCe?r.oWvIenalrureefesarlaitryoe,atahllelowtndeheltrglh-Ferdeiemfisldeo.ntsfeBsdEayQ.lltoF,hvwRuer,rtfiChFfetlh,romaoocfkortreisoeani,nlximdktoheufn.glotarTitdapejhdldoieciy,n[aEwitcnieo:gnFleteia]sl,eridnmwishea0tnilrclatihbasn.budisotTiu1tvh.teetoAhdvredaeimdtrriaiaoettndiioosdaninlio,tanionlunoemg.fuabmEDteiberoasenspr,asiatrmeQeu.alAtivpeblriiyctalctaiotonenvr,eanwniedenetncfioeuldntfeorrπmaanndy possible combinaticoanns hwairtdhlythiemseagoipneeraamtnieyoannbsrbianynccl“hubdioigfnggmerra”at?thieoWmneaaltrinecufsemrwbtiotehrtsohuaetndteh√germ2ee.. TofhEis fioevledr nFa,tuorftaelnlydeexnteontedds [tEhe: F ], which is the dimension of E as a In reality, vatelhlcettohfraresceptaasceleltstohvQree,reRFds,i.eCvtWisslioveoToanwkhrybinlaylelkinkooefrtn.mo-tzTnreoeyreuhrew’osetaloyhsyvmoiadccnialhenoustsn.ehdcttserhoHaimaeibonriaerweot0tesaihcivlfizalieestnlwr,,dietentdhilhtl1ue-fe.udicysoalelAfitabdibndeovuseadncetdri.hrt.eyiteFolhFmaonreubree,rso.sttnarhrTaemiegtchretammeitvoysoioteonsranteioarc,,neontmefuberiutoxteol,tatho–mwip.eiwplrFilrchseoata.ruttiidTeowyxnhaeaicinmalsecknrapdtdnlolialesdfit,anrtftihofihsbereeumtlctdssio.qvpomeufaTapornhrtlvueeeoxmsrrfeobatodehtrdesiotbafiiong2ndg.heoarDpsseefsratpaDelianmtiotifoenuresalyl nProblembuemenbperrsovRe.d rationals and is of deTgrheeeretweoxi(sntsota tnhoevtei2ocntioncra√slpl2ea)dc. efiAeonlvdeerexxFttee.nnsWsiiooennw.oifWlldeneogktrnetoerwytwthoa√ditseQsccarlilbeRde tahqiCsu.aindIrnfautoliltchbeurxttweenloasrbidoosnr,.atthIeteornataionneaxl√anmumplbe.erTshink of the complex division by nonpo-lzabenjreoc. tvWsalteuoegcseatalhlreitraawltlhipwteohlenaflnalt-echdfteebwmfiaelncl.maetBduoh.syrreeeFteiufLhtotresaortietmshfitbesefrtetrtw,hvmhiwoareoredaryfreticsomie,oionxmentonauherlsliml.intgooieroApnaurlaodiesclfjelaaoya(,ltiaisnwizbonieaenstotglsriehnpsaa(eeacadrictrndci)esadstaaniwrlznlinlgebhgoa,eue)tbrteohricvauoe2be.ymtytoTtpob⊂vhohueenQearteyer,aa⊂dwtathirdiemoeintoesisbanoqaltnsumaini.maoiuenrnDpmesaeowtbrsrnoepticoearirtswtneutQforfieefn.reo-l–At1dwo.wadbTtdheihtanhaeytoleasntrwte,oedaenrwl,eQcewa(alelrea2n)ffic.ooerPulcdeln.edtaeTstreohπenesoxaetnedntdheonuerwhofireilzdonis turns out we can caornestcrouncttaQin(e√d2in) atshealprllenaaunlmen.ubWmerbeseocrfastl,lhwiethfiaocrhpmlianan+teubbr√nec2aarauessecaoiantntiadsinbtewvdoaridnyimothveenrcstoihomneparlaetaxisonnauamslspb.aeScreism.wiIlhfaerorlnye,ebfiyelodneisdciomnetnaisnioend we refer to the real the fact all thlriecneaen.sehAtasnrdvdaliyrnydimeenadog,rinm[oCeboejau:ensRwclyty]hs=biictnrhoa2gtndehwcotehhinreorofsetiwnmoztfieoihatt,thettthaehgaelelsemfynereotaewblrtodaieifcmotasnocrcoorawhrtsmlehistomepfhwfoleoserti.autxhmhmteTnteohcuhnoemsemeytmbrfepraie.coxlrtentesuxrtabCrnleoi.eunt–lmhemwboirfhernra(asattbia√wosrnte2ra.a1clcaIatnlnl)doarlffigido.erbrlOmdrt.naop.TtapThhrretheesesoeeytfrhvateherrethhsbaoeingisdgmt,erprutohcsrettutarnoeftornfeoaawl afindeulamdybsweoernsneRee.d to make sure adjoining any irraitnioantionalnaoltshqeur numbersa, rweerocaoltl (aretlnhinoe teal.a tveryhAttanetrd oconvenientaifn(dafieepelder,)f[eeCcxtt :fieldesRnqs]ui=ao forrne2oe wfmany.ght.heer4e)f otreasons.whrmoeueglrde.nleFe roaardtvolvesoetrxosaomafp qt theluheaeCdc ro solutionaimstiapcnlexetxnet une ofnmsis obi twonoen.rso af simultaneousrRe 1anadndsoi.isOR nlinearthe o t equationsher hand, the objects togethreerawl TnituhmearbefeeerwxsiasmtrseoraoecfnafionLtrfiaimhotnaenirtde,hcalewdylleferigmdocroneafimtegelioelinadnvedteeodIraxonirntnfehygrna(e,albsaibriszrloeuisaetnatbnyrol.t,csanrhWca–etlco)elwctfaiatkenmnhlngcnraoe,aowtbnemthrnetsacuomheo.ltatmaTsitfiptphnQiQlcuyeds,tyinReawag,tiRfirChteanhenisoltdosueqCotidunkm.tiauvhpIarminelodimbrkironteeo.targ.hontoeatTrfronhegwfoeo-wyn1rea.dclrdslToa-,adnthytotehesarfinesionn,narewased0teaioaobannordaevelelf1e.nom.urIcnmeAendtbdthesdteroiostnieoxntth,eendengeoawutirofihneo,ldrm,izuholentnipcelicwaetiotnakaenadll Similarly, adjoininogf aQnHowever,. iIrfrEatiiosn aaitnl ciseux btaee rnveryersaoilpoonot surestrictiveoswmifbobFulel,rdcstohlameerabe qdoneiuontefaosittn iaiotoo.fionncsuitwwb eForeiicdtaheer gxetexample,rhteieene⊂nstesoeivroenps⊂r,etter ohdatheretiosor nen“aesHlwhichsioonf–wcdlwumedeg uir cancnceaghenrbn3 a be.iotgtigFo fieno donernradilesnxaEua fim bytnbhpi teal theern,snFau nm? elementaryd”b√eWr2ho. afTtghedinose fiwr operationsaeetldornsaatsuraab lol ayv e fielde.xtInen tdhsethe can hardly i3mraeagsritneeocfoantnhtyaisibnareardtnicinhlet,ohfwetmeorTeawhtatihehlllerenemuusesmaexttbiifioscetfnsrsicswta,odeimwintvehhopxisotilticeiueoohxntnnsintnbi1chouyaentm3lmunslbe,ro.dnetnrh-fias2zaeretCel3rdi.ocseovexnaxttletuaneienssiseoaidonrne.insaWlotlhefwefikenclolio-tmdweepdfitlhengxaertden.Qeu.FmubreRthrse.rmICfo.ornIene, mfioteuhlldteirpiswliccooarntditsoa,nitnhieseddriastrioibnuatlivneumovbeerrasddition. Despite Q(√2) is the set omf aelalsquarennubmyb“e brootrisggoef rtof”h?e 2Wfr oehasrsemtro eafamouslyfe+trhbti2so a/trht+e icbeendcle2,gr/we eprovedwwohfielElreu oastove,ebfi r,bencFita ,irrational.eroeefxrtaetteninodsnieoan nloss.t,allowseStdheva[⊂Eetria: slF⊂ (addition,eax]d,tjewonihnsiiconhng si subtraction,smotfahfiyenbditeiemdeengs rio multiplicationene. of E as a and division) There existsinaFaninoeoltdtiohsnearcl,laowlwledcthafialeelrleatdhccetoixIonltnanettanrtioseenfiaroeladniatd.yij(n,oWfiiatetnrhelhalideenkt)tignfrhoaeeorncxeaweatltleeleastnmnshluaesleamitntosdhtbnQsrQe.ieors,esfRTo,tsfRw,heCtdehseirfcglvoaohCrartoemri.ikeoynIentnaretwa.lunliokroFtnerh(oum.neramororTetebuwxhestcaoelrhomyrysendpitasc2nal,roeieintnhCteaaed√iivnrsi2aens)art0i.yintzohaeAnce,naoxndtclnthoevn1emxneuy.tsnmpeiiboAlnebenensdxeaitrdornsniffitueiRhmoledfnba,dsenfaeorndmgsre.rgsemeoIaefstaitiotnsorwnyinRceo,tufiimsreeulcdla–tliilspweldchiocaanttqiawouineaedcadranaltdlica efixetledn.siTohne.seIt vector space over F . WFe iwelidllsnQaoltlotwrytthoe daecstciorinbeofthaidsjioninQfiunl√gl beuletmeleanbtos.ratTehoenrantioenxaQlmn√pulem.√bTerhsinakreofathvercyomcopnlveexnient field for many perforrmeoafesdQoni.nsI.sfeHEqouiwesneacvneeAdjoining.re,Fixniottedrainisenvsxoiaosat inhm(adding)voenoprfl,ybeFyw,rie,nfsot cwthnraoeti-lb2czultqtje rtaeiutornvhckoesetes svotolaiuatnolot,uen wtgweeweetswrtoemhae oacobtainre.aer(yn⊂rfiFaweeoficllilirortdnwsnh e)t⊂tsaexatel ralrnewxen-fmudstelctaewepetnrfild gseQnmfieldie,eio(sotdn√rho.“e2eoH F)denotedfsuo(atqwrhsumteh2ame)lrtfleurohanmcrenrumh odfmoreboetwithoutrbih,.ngeoemtgrFfnesluo2irofnrlutifehsierptatx ohElshafi takingecemf(tarfaahotpmtbiralomsenont urCFa squareis(slc+?yidts”)b2ibs√a,Waetnlre2g ihneb3ea rootsax)buspt.trtraeiadovn.aoves neTiwo ordhvneeb ryo anythingvfaadRrrydeiaostnvoiodenirm s.t fancier.ophDoeiesrrstRapatinti ote Onnnaolw s. theaSdiam ysilaornley, are contained in the real npulmanbee.rsW, wehciachll iint atuprnlanareebceocnatuasieneidt iisn ttwhoe cdoimepnlesxionualmabseras.spIaf coenewfihelrde ibsyconteaidniemdension we refer to the real SuppotsomebeFaenisrbraaytfii“oebnlidag,lg.Ee(rA”aod?fj).fiotW QnhPleaseinie.etienIfrfagecExf(tet arnoteeaidsntldlosairtnoteh gnhtheare)esaexooe√dtfdn ejenewsi2osntge.isrtnaeisoHoinen nvdQofieldgoawo,fLraefywEnvaFe yeisorn,fio ,ivontnotbreihrittrmaeaetiiF osqinalleou,xanaseot alsverationalsyfnltneeiersiownyqinoudtfinrwaheerosenledtifora rEtwithidrcoseee.tidozniitvTeno[ e,Et(honeentoerhd:oneetFesQ y[tLt]et(,hbod√other:waeotF2ia.hs)rio].F“cft=hoHP hhand,areoli[esLwepasxtaes:hamremEmfe euthene]cdpco[tEshiltmete sr,latterb:qeitticuFnhghtasgeue]ir.eronesre nTeqtwoiew–uhso.gaifEwsfi.rE ehconstructions4tla)drhatoaswiownsaoteuFolcfd?a”2llleWhaadhsfi mayatefotaldmd.ao oTinvolvequwuhselaeysderbae ttakingeinc epxrto evnesdion. in another, we call the latter a (2field) extensCion Rcoafnthhearfodrlymiemr.agFionreeaxnaymbprlaenCchisofanmaexthtenmsaiotnicsofwRithaonudt stoheims .R demonstrates the ldineger.eAe nadctisnidneeadp, r[edtoi:ctba]eb=ilrer2aw√twaioyhneuarWpel.oHtnAYhedmCjgoAueiNlnt’ieiTpnrlgaWet(oEearxds.t.deo.ni?fnstgiho)ne√c(2oamdtojpoiQlnex,inwngeu).mobbtearisnaarene1wanfideldi. dOenottheed oQth(3√er2h).anPdle,atsheenote the new field is of Q. If E is annvoeetcxattolelrnrsaipotainocneoafolvFsextraew,rtmiFhtheo.a boqWelementnjnueebecystwetsxi“iotltblonrigang weogetetlh2ertae”.mrS r?InyieemwnTWt iinoordertihtlteahederrreal2eeys.fsc,f eetertoxIarewinibdst ejtopreservemooissrtitondhah“rieHensrgofditontweoairgo mfnrpmuthenerleiutlecrhcrsabo ahelstructureuflrteftvbiErdoeioenglfinotgahvteelblerldocrinursiesFat ebxroftEe,utoeefco rnotafo(nhtsu aoeiarbontaennssquarenwtdo.FrefoaWx?nuac”aoltedm)fitWek earootspldnleldlhgoae[aewEd.wtbTetr :doa(sincehFno.aiean]twTe,kcdQehwuoe thbcirclesfyoitcahmRreeaixsckts oeotareCmnhsi.semupi Iorldgivenpneenixom,ortoehanr enbysoritonw nequadraticoowrfdasEd,eatgayhrsee aoeequations),rna3et.ioFnoarl neux mambeprles, real numbers are of innofitnaitlel3 rdaetgiorenealosvweritthheonreaelsxt–rawelceamnennott√fin2.d Ianfionridte1/rn3tuompbr2ee/rs3eorfvgeetn⊂heerastt⊂orrusctausraeboofvae.fiIenldthwee need to make sure mean by “biggaepdrld”ai?nnegW., eWsurebetfceraarlclttfieldoiitntvgahe,ec pmwetadloanuerngl htesrneed€iepapeb raldeycocleiyf natooguEivms eaomakeranvQigrdFteei(rni.d√cseFi oWv2tsurean,wi)netdooaiyifswnit ndbtegiadding,lehinrlmdaendnreisoeencetnhstwtiotoh ertsubtracting,nfefyllad-mrtdleol[eaaEnfitdlshunen:aemesuFmdcsbmr]paie,obtarn wiescmultiplyingcrehseotsilhf,wceithwmshihtheeihsnrinoceftohufshubrteimlyo nltndhobatienbututm+mhertene.bedn 2laninothingsamireboeweon+ncrofisacintfeo2etlEn daelse.o,iwnaheswaednahr eiceHence,nefrexetrwahamte,obptc, talocehwekm.eaepTr raelmayehelarlxianl ntki uoonlyomnfabtlhes .readjoinsS.ceoIvmfeorp nalsquareelxafidejlodi nirootssincgosnm taainyebde linTe.hiAnnkdofinroednseete’dso,fm[tCahti:shReamr]t=aictli2ec,awlwhaeederdwueicinltalght,uieossngeue.bfinTFtneroirhatracettoiehSnrxeosgt,melfionufemosltdsiuhotiloentoxinspcft,oelitytnmh,isnapwigotlenixassntneduxudtmdyeinbvcseiedirorsitn√aasgirnoeafsr1efitaQnQswniteodelfli-dnR.deueOgmfirnebCeet.drhseoanontdehleeormphearnanttsdi,ooQntnhs√ethe new field, hence weQta√ke a√ll vector space opveorssFib.leW€c eomwbililnnaandotitponl tadividingrsnywet.TiotWhdeet erarehsecereasi xelbwell-definedlieositptitsnpehaareiasrpnftoliioanor3tnmnih.feousenlribdl n,onecbciwcanlualue ltuelementssedecsdeiqaenlaulglibetrtnoahircsteaeit.olw enaonFotaootlnr edthenrieaumxanmae .(enewmnbfixWsepeairollmesden ,)field,akpainlefedoxa.wwtsTe nat2hthsa.itosiankopTtkena hcooiesff. wtfiSimilarly,thheheleemdrcfeanoymarbmtyfip.uerl rIoesrifantxn.l leweeoFyntodhelria xemarertregexwena doworkingstmiorsodpntslhe,(wetCeh2e ri)inseraf aethentnridotento xhaFtt leeh-plane,nneusfiumroernbathleo reonefsrRto amaynd( s 2o,is 3R). Fields allow the paocstsioibnleocfoamdbjoiinnaitnigoneslewmitehntsh.esTe hope erraattiioonnasliQnculmudbienrgs⊂raartei⊂oanavlenruymcbonervsenainednt√fi2e.ldThfoisr fimeladnynaturally extends the plane. We calorlanriettaiohalnenpamulasm. naBbeneydbrstehicasearouefifsfloeitdfnheaieitgnro.refiireAsncesntoitxwndewttohiaodndiegnd(igrnmeradoeededtenieo,nSt,osf[huivwCtQoepehner.p:e2aloRtIelrihfsaen]aeEras=nl√Fraeina2sauils)slawmp.sanah–bAbcefieoenwrxreuestlewetd,etxchn,whttaehsEeehnirenoeigncnasreohibanotfioytneinfinfirooiFanontnttedfau,eoldretrandhxesgifiuetomaernmfrqeneieubtsnheiecsotesorwintosn3uciontoomanfimw.isbnonlyiwtepAecelreadrelbneoxali feidfnradjoinetdnergLluteaihtmntoaeteqberurctfi er,ahsquareoanrewtmedsiotrterpaeesreldteaeinacxli cs1stroots.onea“uaxnubHtnsmeotdionovebwen srieThat.irπoomsIOnf.nau.nEnIct fIhdis,.ttoheTb neifiehg oexgfitn eheisrle[d Lriansih:sE aFelementcnot]dhn=,attan[hiLneFe: ?dofE” ] [WFE andh:aFt ]d.oTwheis At this point, wreeashhenceoanvse. pH rwe€ovw ie dtakeveedr,e inalltroaiust gpossibleioahnvbaealrscykagnr erdcombinationsostuirsnicdotfiovdneegofi4rneeleed twithttowhooe.o (rFthesenyoortoetxhp aeroperamo2vpeilne√,-√t2h2ie)s.snqAoutnacreoexntrseotnorsuticootnfib2olefhdsaoesAglvfTraieAnmeNgoAtwuSsoAlyTisAbNceaAelnSleOdpVraovqeudadratic extension. It line. And indeeteurdwer,snht[siCcoohfu:tdtRhow]is=enaocr2tatnifiwctclheoret,nehrwaseientlertonuhawludnceimtolglstQbechuhne(esr√eresmr,2afiawe)tnr.oeaimrtsTdsoceeafhoaemleiflxnyontfinhebuannsyemritsetrci“lbeoaobentdmiortesgesetsp,ghrolterefhaerixrt”eaerh(?tnaofideuvtieWsfimelgodrernb)xemtaerehtelereexasfanetc+rnasreteidrnsobateso√nlfiis1nos2tr–nhoamaefnwospdfifdeparneateicgaih.dratrnteineOcdeontfdnoafobeorbtgtfvmhlrafieEernerwybod.oatoivgyFahvegoerfiuernrpFtheiohst,xaneeanotmfdnrtoau,epuftmntillrhoetbedinepaCealrnelsnios.eutfxeaSmgdtniembn[eEiesxrlaristoa:renRltFnyo.s,(r]ia,sodwnajshooiafncbhiRnoigvsa)en.t.hdIensodtihmiseeRnsion of E as a the Delian problemto. btionse irra tincludingional. A drationaljtouinrninsgo u(numbersatdwdeincga)n √andco2nt sotr2Qu.c, ThistwQe(o √fieldb2t)aian snaturallyaalnl enwumfibe ledrthesd4oefn toFh t-planeedfoQrm(√ ais2+ )constructible,.bP√l2eaasseanaontedt bhsovea nrisye wtheovfiee rlFdt h(ies rxa)-plane.tiAoTnAaNlsA. SCareSAimTAil-NarAlSyO, V real numbers aLardaejtoeoiFrfn,iiewnlfidgenscaiotanemlylodeiwertgroartehtrrieeosoanotaflvaicozQeltferis.otqtohnIhunfiesaeoErfraceeraaiatsrndlioscnjaolvon–etiet,ncew(iwctxneo2teogmrectwnaspestnpilulihenalotmaconeutetestoneofihftfvnefisFead.nsr,iqaptFTtueehfiha.renrefWxeiqctruretaonetnwssoisqoutitioiulnmolonanfnbsrl,oe-wn1rteu.hot.magarTfty.rbghei4etesr)oinnnsee,wdtxraeweatorsetueecnoslradrstaisierboedlvaeenesfasitorsdhayro“ibcfstHecofidoivonnneawti.vftouqeImnuleldxiuaebtetdcnhgeuhrtnertaedbtfieiie.lcgoalgudbeerxorftrhoeiarosntresEmiizoaotnnh.yan eFx?a”mWpleh.atTdhoinwk eof the complex If F if a field, bnyotthaellFra-pt4il4oanaelswwe imthaedaojnonieFnienx,gttrhaaanteyliesimrareallnttipoo√nitana2ltk.ssiInq(nxug,aosyrqre)duesraurorceothtorAot(ApThonTrAtaoeAstNsNe(AbtAsrShoivSnateAchAtTetTfulAhxoAcNefiNarAa sAncinspectionStldSrpOeOuseVyVcratfareurecretegeisovlq efeofumnaa ebrthefineyetlseqd .constructionuogwfa. dFe4r).natewiecoduelqtdoualm erulestaiaodknest) osdemonstrates,uabrueqtunaodtrhaitnicg e therexlstee.nHs ieonc.e, we may only adjoin rest of this artSiorciemlteahi,sleoawnrselsey.tw, Hoialfdlojcwuooseimenvextendsiepfinrlng,eixitamteFnieeusiaeix mrnthelatrdebasbvnteeys irarationalsroisloy“nlnoCbarwsil.eg,scgttpuehrlbriaec”e nttand?aieirvc.sWoetoei Wisoextn teofrweeoncof tafesdegreeuorillaoodtdn.iotjsloFeatoiaohnf drpetwoifinletadngxoneiagte ea(notemrlebedcepmeuelceogb eathe,fnriucetEsehse. e2xoi tvstTineqehni rusesaFit2orrwe,n).ao o,trifoodtorenimtnaoloendnfnesu2nTioomfhntbaedaedslerSgfas[aorEsemalear:uoe3Ftus.pasi]ola,yFvcnwoebhrewyiechxhnceaoripmnservpotbehvlnyeei,deodnnitme fiedenilmsdioefnosroiofmnEawnaeysraefer to the real 3 Q 3. Can wetoc√Iobnnesriterrarulaicttiyoa,andalaldlllip.ntoAghi,ndretsjsueoibonstneirntastgchtQ(eian,dgRd,-S,ipmnCilmgau)cliloleta√oirp(k2ltlyhy,atieolnaikgrdQaejaot,.iniwonTdnei1nha/do3gleibyvpatilandcs4aiqoinni2uner/gtar)3a?ranietnrieWorewo0newoafiatlehesnlaclldt-uvdoebd1eQefie.dni.rsAoecStdoduietmdsosiwnietQlidaooe(arerunl√alelyt,dma,2s nke)noilefe.gingamwP tgadfurtherseeltesniotaqoaotnrnuseeo,atfwrnhmeca oeupossibilities.unrtbnoelkytieoiictnwrpthagsleeitxfic(AinsatoeTieetnlnAdtniwaochN,sleneiAfih ocNoticeSeFnailnr,And-ccpTdoleiAlersasNwno aAenthatrS,eetOoagoVnkif eve[Fdem ane(lgalbryyexeo)qn 3u:l .yFa da]F rdo=a jrto1i eicnorxeasq mq2uu apatlreioe,nrso)o, tbsu. tTnhoatthins,gifelxsei.s Hence, we may only adjoin Fields allowQ(the2)aicsttiohne soeftAnaodfrj eaoextensionalvilnseonicnutgosm.rbesHlepero masofwceoe nfvdegreeotetsvlhri.3e,nereiTft.FohirA .semtwonWardaaevt+ ieiisnworbd ynicalled2lealerlnedsno,+ttu[rCm ictcar2t b:yquadraticievRtre]oswo=danhere2esrctwearo ihaoextension.bve,.eerbreF,tyhcotirhacseoreienxngvarefea mnuntelipilroelanbet,atuoltrsfihs.eelloSadfsebqvtfouhoerraeaartmcleeoaarmodonnjpoyotlaiennoxifnengx2usamhmabpsaelyrfeas.bmaeTroheuisn1lkyanobfdeteihn.eOpcrnoomvtehpdeleoxther hand, the length may be conpsotrsusicbtleedc.oItmfawbkienaghtasivoqenuQsat(orw√ecitr2oho)nosittsthsrteu(hscseeitnosacpeeteprcoaifirtncaitollelniasnsnauitrnmehelceeblgmueridarvesteinninotgfnbotrayfhalFteqpiuofloanrndamedrl,antatwuhi+cmeeembFq2eau-s1rpyq/as3lutas+iatnoranecder2sti√)r2s,ow/2c3obio.tuwnshTthstnhsteoreoirusgQetmchfia.tieei,nbSlnbdgli,temscen,liaoslatoefrue.rilrsyrHaa,ltetlihnyifoecweneFxea,tl(waes√.n3reedxSms)we-avtpoyherlraekoaninlnelag.ydCiajnodaitrjnoheiifennugFlsi-nmpslapayencebt,eioneomf tahyeocnolnystardujcotinionsquare roots. That is, if x is dpnievoriftsoiroamnll ebrdaytininoonsnea-qlzsuerwetonoitcvphbealt.ealauonFiknereosier.nraagetWerxixesoteqanramuaalclalpe.wrlllereAeim,lrtadloi-lefjdaooAnwetitputsnfielmin√(tantshenbag2idin€eks..(cre abpesIFQndeouaccdiironawrietrntcuhed,lgosee)ewfmrsrmi√eiantyrofih2eirnasfieptgviro,trieswevmtQsepoduen,rrenlodwtvgblviearpyemierdltogeqihecbenudoaetsatvaiesodieoiotnrnrQnoauataut(hilcis√gctenahud2esrriw)qesb(dependingeaatuar4filoascnsifpebtkd–ilaugodtcrtnwhfioeidsveue)wenln,cdohdbaof veuntwwhetheroerrnterdntonhanbtQfioedeyrefit(edhl√ndtoiitodn2i toexgQ)oahn. de(is.mfiPo√li snmDrale2yie k.te,asquareesnt√Hsposensie3utopn)ernm.roceoetb w€ve,ineew tr hFero e).fm2f negIfariesy wntwenoeofiorntate lhstarttlycdoeoaArirnsdTsesjAa towithsNlrinuAacbSto iAbthevTleAe.N sIAonlSvtOihVneg reasons.leHnogwthevceorr,rietspisonadvinergyIttor eturnsestarcihct oioutvfeito swenceo tocanorod.i nconstructFaoterseaxnadm tphle(,ntuh2se)e asstq√h uallearc eonumbersorrodoitnaotfe 2ofahx athees tf aomoobutasliyn btheendepsriorvededpoinqtu.adratic extension thaeddfaincgt,aslul btrhtarrtaeiceotnsinaelgtssl,ianvmssneqaqu.duruylaAtiariserpneneoldyrofroiriondnomopgetdtgoseesaruettfentsodheold,ryeQmtQ[dwCDi.ne.iovedS:Slitii(dRihmanmien]oiisli=gtralpearqtsrar2lhiouylrzybe,wee,nl,i2hewficftmeewihrwtn.l.euale-yFkldtaeoaihesrbn2refi√ge)den.awgexswermeqAoadonutrmonrehkaonkrpeirneaniselnxtsegergotar,aleoirmeinitnsfonmeestwotsistehofsehtn(etnhtetsrthesFiaoFncerkfot-ece-cpunedpaolrQealrcmetagenih,nr rw–pewecnel,eleo,wenolxmos ethfnmayuwnmwanearerotuyetmefimhnwifi egstaeonlybraelirydscovyetcpa,feroaoslnoenFhll neslalachieveebsyldlnryaiyebcnarafieqigd1ldeuijwtjloatidothiendo.ni senextensionsdr.tQaTsaFsiqtNkh(.qi-u√ceoupOastale2areianqrc)elnelueatrearhtonotiheods ioaFototc tsnsuchstho.s.[hneF)TenT,.sr(htbh√Ifr ahutthatuataxtrtcnt)itnsihids:,boe ,thereFtlirefihtf]t,hxioxn=esogiQsi 1seexistsi(sl√ostre2h.2,eH√( Fde3ne()√pc. ex,n)wd-pienlagmnwaey.heoCtnahlreyerfauxdljioisninaspection of the construction to be irrationaSl.upApdojsoeinFinigs (aadfiformdelindno,gt )Eaa√ la+l2 fibrtanotiitQoe n,aseawxl steaerw noeandsbsititohaon ifobnont fahvaryeinstexeawt rnrovertdafiiceeLlleld e,athemwdfieenn witrationals.otteileled2ux.Qst€eI(n √fis noi2oSimilarly,ir)tnd.eeoPrefxlteEtoae.snpe Tsrineohsone2etnsre,v[teLhtea:htnFeie]swt=erxfiu[tecLeltdnu:sriEieso]no[Esf ao:fFfifie]nl.ditTewhdeisengereede.to make sure Could we construurcnts poouitnwtaasnenoceuealeltnesm2micdeoeenntStsthutoroepfufpFrcFaottsaQieaonn(Fnd√datl2iths)hpeaelaaFsfiFnae-e-pl?pldlaln,aTnuEnshemeqaaiubtsisaefiicrscnoesointonmirs-enfosptptotrelFraohutxunsec)tcs.tetfitinbobiIbsfrlleQimelwoe,u.n,esassSoois+onitfmiagbsiisr√tiotltahn2hwrnelelaydiryFtsu,FhtLal(ihe(f√te√sahwnxdsexfiedt)e)rn-Qma-pabiprt,liovealgewananhwnersetexyteo.rme.troaedCkCvatngiaeynsaesrigreoatefnihlfunouleloynlrtfirehniaan.Eecsatsphir.pFoeieTnc-nvcpathteoiloiesaoen.fnnuxSe[rotL,oitefmhfont:enstihlFrhieaeoepr]nmclo=csyoaso,nFsyn[isLbstostrinu:rluiulcEtcyhicte]tia[soitEdo.hnnjaNo:tiFontt]hi.sceqeTruethaheirasextirs[otFso(ta√s.pxTo)sh:itaFitv]eis=,ini1tfexogreirs2 (depending whether x is a not all rationaodlbpesmojwescositinbshslteoroancgtoeemtsehxbettihrnreawtdaieiteodlhegnrdmreasieaneewflgneni,atwtuchs√tmumstb2bhoit.nersreaIaesncftoapoirpnrrmeegddr,oFeaItirfmicfhteititFelnouaodfinblfstripnslfioepariinatenlweyltscofieaildwneruiyenlvgdgeuti,rahnptoebneghoefydner(aoatcdvsmhtbteiieivrsuroutinlFdtrtcahitpnolceultgfnrer)euaeaeamdorxlljegfsbtoweaeiw–bnreerfisiwnlaimelaoe.g-lnedcTadea(wnfihlan√enednFmye2joandoe.t ae,inpositivenTreofitedinshn.saidgtseoto)Tlafi.eimshefi leaintegerpdnelknoirlnteatrpeastsoatinuouinunrrt metnasnt lbhlando(ynexwrue,anmxy odaetf)baw echaingsyenuersdfinscseohelantrd rthaeof,ehteoha a€rteintermediatesn vbcaeoesrtyhawbcexotvanaenvk.edenI extensionsnyiaelantlhrtefieelledmfe onrtsmoafnFy. However, we haveandojotadjoininginuisnegd athn eyanycirorm a irrationaltpiaosnsaslosqfau r squarea.reItroiso t rootim(neo tt (notonthfaoan rt thatmdoafllac yh ofpaei snrs afe eos perfectcftaidnsqdteuinramg reendeiw.atgta.epk4oien)ixngwttesonqutsuolidaornaelsecrahodaorttso. (aTsihnqicuseadcirractliecs eaxrteegnisvieonn.by quadratic equations), but nothing else. Hence, we may only adjoin carantihoanradllsyainmdagisinoepf oadrsnesyrgsirubtrubeloelreesfasctntdowdhcemeohimsmb(oaidnonfnrenoatsmmrtsitCectriaotarolaethansntnh,oetsesewtnswm2swrsetae.tiahitthwntecHheisrioc√rloetnlsetwh2hsauweate)erssr.vieeutdehAncfireonot,gnpouriiaefatetfurlneulxairtsrthpeatehixleaochnemtinemsrnevsrsi.nteopeispirsnnoyiootscaossnlrosfiuespbifbdts,rihtFlietrilnehgdtiitgciariQeictentrsites-av.daip.testbNlitwNaoleohoecnoxneteatweiieFcils(canettnes-yocshpqutaiotehoulmhl.aalnparaenbtsFraodteetno[orF[iiarFifsosnm(nefiqa(c√uxa√Fnolaxidnt)xtmdp).ise)p√rtl:pIalr:dfe2FluntFe.iwcegc,]t])Txer=ti?e=tbhsxelet1iW.tne1sae,sornfieoiqstroserouhni2wloa2daiinsr(vt(naed.hetddaehreIptdjoetpouheioiesrFnetcnadiu(Qdlonil√sniyfg,sngx)e2weg.d)xwe-hwtpahaemhlnseaseatdnefthyagsheem.eotrrhnCoexexlnuaytisrsilesayoafcfauhblaieenienvynseprpeaexrctotitoevionendsailoonf sthFe scuocnhsttrhuactitohnere exists a positive integer adding, msuabytroaccctuinrgo,nmlyulutnipdleyrisquarentgheanfod le.g.lodwiv i4)indg iwouldncgiracurem leadwstealnl- €cdtoe esfi a(n pequadraticrdovoindeedleim nextension.teenrtssecotniotnh sSimilarly,ednoeowccfiuer l)d., hseqnucaerewreotoatksetoalQl . Similarly, if we are working in the F -plane, one may only adjoin square roots. That is, if x is There exiSstims ailanrolyt,iosasnqdqucjuaoaarilrnelQeindini√gnfiFeaFln)d)..lieIrInfxrfgtawewtnehieossnmistoatanalry.rtctuWbwwbeieitecthakhordnnotjthosohrwtetiuernluQwtieQnchso,tg,audewtwdleedmQ.emlImoefnaa√wysdyRterotoahnontalelyaCvysea.ctacthuIchonebhiriecocevovteaneherxsenetetrxrexnatuwnteoncesontdnifrsousdainaiorso,tpn,cnhoshtseihraFnrQFeiotnpnsriosuaoneustcfscithoioh0bnfnteittdhalehirelatragimtnetr1usiteot.hmdehneNibea3rrole.teertpesiFelcxeaxoexinrist−stteethes,1nsxawasatieomp[pFnomposs(lsai√eity,tixivvse)tea:iinrFntte]weg=geietrhr1 oserg2m(ednetpseonfding whether x is a possible combintautrinosnosuwtitwhetcha3ensecroaontpiseotrnraFuaticliesotlndasn(dianlci2lsol)uwotadofsitndbhagelelgrirnareurcteaitmoittonibwonaenolraosn(lfTnu.oofmhAtebdhtejhSorefsioonal2rinmuigdtn1eai(√lo√+a3enm2d2b.d)e.iTnn2tA2hgsa)i.n3ss√fiaeTx2ehaltdtneondnrsQabiot,viunoawarnroaneyaflloeoydblvneeteumgaxrmritenetenbhentaedtrownrsfaeot=atFwhirioeseFfinacenaallddslvl.etdFhdSreyeinmaoFciqtol-euanp dravlldaeyQFrn,an(ei√teiincs2t)ce.fioxentPFelsdnltersa=fuiosocernFtni.m,boIlateten, ysthoeisntehwefiFe(ld√ixs)-plane. Careful inspection of the construction Q(√adjoining2)€i s thne aansnedt irrationaloaf cahllaninumo fcubebienrtse orrootmf te3hd.e iawouldftoermexat elead+nsbi2o n/tos +⊂ ac 2cubic⊂/ w exher-e a, b, c are rQationals. Se⊂veral ⊂ad·jo·i·nings m⊂ay be (1a)aredAjcdoodinitninagignaeandnyiinntiretrhaseeticroteninaoalsnlaonnspuqdsom.uianbaHtreceorhoswafr,oeitwnvwoleehtonrif(,cglnhitintohitetniescsr.totmharuQarrevetndes√iporayfortneardecepsioxntsentrqgretiufcnaetatioscinritvoeeendsaiqsncoiuhnFaeort)fhet.oeietIofs.c.gowc.Fmoe4oop)rsrltde√weaxixrnotanuamwutldempistTllhbe√ah,eanterdtdhshe.StetoIhofselqa,onuuwnqtaueeurisofieamdenrtalrohdyaoetitoiscnoclefoyxrn2tdatecihanhiansaisieteovedfnaea.mexxoetusestnloysiQobbne=tseanFFinp0srtuhocvehFedd1tehsairtedthFperoe−in1etx.istFs a=poFs,itive integer rationals and is of degree two (ntoutrnthseou2tiwne√c2a)n.ncoAotnnsatelrlxurtcaetntisoino(nal2so)fwadisteshguarclolehneteuthmweaxobttefrirsoasrceoaelfaelltmcehhden1aftoqrumia2d.a3r.+Ianrn3tbu,icolFer2sedxdieatsrsemanatosodainoepngnsrdtr.erebsaeIvtrteavwsreyotthoheevxeretesretantrrhsueiecontnruoarotfeifuoForntfah−laes1.rfi(Spefoliodmrsmsiwilbaerlillnyite,i[eFds. t:NoFomt−iac1ek]ne=thsa2ut)r.e[FBn(y√mxu)l:tiFpl]ic=at1ivoirty2 (depending whether x is a (2i)n AaAndotdtithnhegirs,apnwpoeeinrctftension,aoe,lrwlsmeteechdtheiaoilvn aoretsptpe eoqoneriounavetin (doofcfiefeed.a ldegreedeFl)inonoerCexueotagxeunh anl3.dmdsb iapoForwcnlceki,g orcexample,icrfofolntewuh.sneterndtufaocoak√rtnme dpfiQeoeari(l.nwd3ctQFhestoahm)ori enisuaoet yoxrstheyfiafidmirtenso pttsettelph≤errenm oClrvae≤aderitsgii√aeoatnt2neaoeileisQxptnt(leea√onnts2seici)?oonanTnssodhtfraQtutRhceit√sanibnimfldueprstoohslevsisriibntRlgoe Qus(i√nig2,o√nl3y).the ⊂straigi⊂ht·e·id· ge alo⊂ne. turns out we caSnimcoinlasrtlryu,catdQjo(i√ni2na)gdatjosainanblielinrnrigruartmaainotbyinoearnilsraracola.duftdbAitohiendegjarofo,loionsrtumqinbuwgtaorr(+aueaclbddtr√dionlio2engtaog,adf)(smndatoueota2lgtnraiQttepQdhoecla=ysbu=t2ibwvn,FoiaFgecwf0r0ycaeaoxnoonptdvFbecFe1tnldr1uasfiteidivnohceintdea,irsnnaqoFgFeturnsiwnaoq−o−rnunfi1e1aaeelewrlsdo.e.geFfFli.dSlnd-eni4dmenF=)ge=oifir)lwtFae.neFoer,deIl,ufdy3l,dw.o(enFlesoae2trdla)e.remtxPoaewlnmeaitatpshqsleuoet,anhdo3etrthaQeet,tichnweeeewxnmteefiwaneyslfidioe,nldhl.yeinsaccehiweveetaexketeanlslions F such that there exists a positive integer (3) Ad3 ding anSuinptperosseecFtioins paofiineltds,oEf tHAawotfiowtncheiivtirseecrlpee,xoswt.ineent,hs1iawo/vn3eehonafovitte2/uap3snredodvLitdhaeedficen⊂o⊂nimtoepueagxs⊂h⊂steb·sn·a·o·s·ci·suchfkoagnsrru.ocIu fhthatt⊂nE⊂tidhs. aotT tniformhffioeen reeachlted[oLatcfho: reFm1o]ray=llity[oLaps:nrsEoe,sv]F[eEai d √iis:ds2 Fiani ]gsdegreed. enTgoehrwteisecpo ttwoonwisnottres uxextensionctttoeinbasleicohsnoalrov tfi.nFTgi−h1is(formally [Fi : Fi−1] = 2). By multiplicativity othfQeQ(√D. e2Ifl)iaEisntipshreaonsbelofetexmo Stallfe.imnan lsonumbersililtoannruallmyol,fbraFedtr ,isjofootonhi nfatheepitlnshqoguesw formseafisobtnthrliemoiorn crnaoawem+tieeobbxani2ntraraelaticine+oultnebecsmer2ewersniottetohwherewdt√thiwhs2eeo.r“seuHeI lnaa,do, wpobb,ler, dmacrcade autrarerictoethoon rbsapitigrcnieogucsnebleuraircdlvissie.enExgSttheertevanhetsaritioanQornlunFa=c,dlt?ounj”Froru0ieWomnnoib≤nhefegFaaorst1sf≤fimdadeoanledydgwrbwe√eFee2.3n−.eT1ehFdiostrFofieemxl=daamknFepa,tlseuu,rraellny extends the adjoining any irrational square rootssu(uncchohttththhaaatttffoorfreaeaacpchher11fect2iisqunan,r,eFFe.igiss.aa4d)deweggorreueeledttwwleooaedexxttoeennassiioqonunaoodnffrFaFdnte[−iFd−gc1r1ae:(x(efQcfostoher]rwamn=misenaiao[lcFlonolyfny.n[ic:F[nFlFtued0:r:]emFF=e−−d[1Fi1]a]n=t=e: 2Fe2)xn).t.−eB1Bny]syiommnu[sunFllt1tiip:plFliicc0aa]tti=ivviit2tyy. mean by “bidggemero”n?sWtra3eteresfte-hprleatondettghhreemedDeagecyrltieasoencioncpufarrEopobnorlevlymdeir.uctFniaid,beolrefttwehnaeydfoeulnploowtn1ei/dnm3g[uEcltiri:pc2Fu/le3m],eswxtoftaheni ncchseiisoiisn(pt (hr(formallyaoedvjdiodiimendiennigns )ti.oenir⊂iseocftiEio⊂nas·s·da·o o.c Bycu⊂r multiplicativity). of de- SimilarlyC, oamdjpouipnteiinIrfgoFrtamhniefeidairnrifitanetrlisdoserationals.e,nqcabutQlyieonc(atnu√chdebped2.eo)iFFn irinSeveralogsotr,othosetuxfewbatsmotwer wutoadjoiningsaplreocdlaletfimt,inlaineoielfgalsna,dwnaimnluetsF≤vmo≤u toamayalbaln,tkvei≤dtepe≤crhsluQi yasbebotithniwfoci egftseperformedhedsamxeoelntllgfadeuopryntreodismefioi3iinronvtsatnwitsdo,+o ief(innonxb(trg 2lnw,sequence.ayoaor)ntrgeses+tiuhmwtoceofuhe2l2Qdllt-eah(di g2n√aerwetfie2√ohneb)u2eoa3sr)dtn.ehldioaAnFnx,teonbharae,erelncenxdexmatqfareuyeuemnrnaastrtptirhasoileteonoir,neonlsotneofamtwhldQshee.ni(gc√tSnrh3se2vweo,ef√trfiwFa3elo.)lda.id,sjhcoeainl·lice·ned·gswamequataykdbereaatillc extension. It n ooffddeeggrreeeesswweeccoonncclluuddee IsfuQch(-√tphl2aa)ntewfoerreeaccohns1trucitiblen,, tFhiisiswaoudledgrimeeptlwy o[Qe(x√te2n)si:oQn][oF=f F:3Qi−i]s1=Qa(fp[=oFornmwFe0:arlFlyo0f]F[=F21iw[:FhFnici−:hF1i]sn=−c1le2]a).rlFBy[Fyfa1=mls:eFuF.l,0tTi]p=hliics2ayt.ievlidtys 3 vCeacntowr espcaocnestorvuecrtFa.llWpoeinwtisllonnotthter1yI/Qf3tF-opdliafecs2aec/r3fi(itebhlede,trhbaiystitoinhneafulFlpllbaunte)e?lwaWbeoemrahetaenvoeFndai,snctgreesuhesaxstaemdis wepaallesl .econcludepgTomhineintnskt(√oxf , ytah)neysucrocahmtitpohnleaaxtl both x and y are elements onf F . n Q √ can beSduopnpeosbeyFthiseaeForfileepml deexample,epr,nfoEotsarsmraiybefildeonp ciiftnoee rmsweaeebtxqiito unetakeuneanrsntsniciosao(e n1n.owefi)suFoewtAo fldmaywriidttehdaxaicltnal ahofirstmgdnwespaLcsenloe€ (onaenlarge,aip snidfiteftrdnrewuairtcestietieootcnQe anttox,kis(ote√s einnunQ2cbps)(litwouoaridnaesnictanm)ot lgfoiandlaofEnyr≤natu,.wfi tmirmToo≤sbnhtuleiaelenrtnlsniepnlso[aulL.firmctg:a3hebtFeieto]rofnos=Qrama[n(nL√dad:2+)Edb2ia]v√.[nEiTns2dnioh:atnihsFs)ea]fin.aeTflnuddhrnitbshavetarurtyaololQyve(e√rxt2he,ne√dr3⊂sa)tt.ihoen3 a⊂ls·.·S·imilarl⊂·y·,· ( 2) is the set of all numbers of the form a+b2 +c2 where a[,[FaoFbf,c:c:doQQeang]rt]er=a=ereda[s[FitFciwntonieno::anFcFlos0a20n.]n]c=Sd=luew[vd[FeFenrnah:la:FavFdnen−j−so1oi1]nl]vinedg[s[FtFhm11ea::DyFF0eb0]le]i=a=n22pr.o. blem. It all boils down to the fact 3 is not a power of 2! That Can we construct all points oTnhteheSQol-pultaicoen(theIrfaQti(o√na2l)pwlaenree)c?onWsteruhcatvibeled,istchuissswedoualdseimgmpelynt[Qo(f√an2y) :rQat]io=na3l−is1 a power of 2 whic−h1is clearly false. This yields withopuledltnaengtmatehk.oinnmWsgtaerysacqtbaueelsalrctiethoneraosdtoprertlugascrtneotieoerdnba.aecnlcIts3yfsatauwhinsieednahgiitaspfvarioseenfdtc(dwioi2ce)otrcga.orAdbneOiledsmetndtrewwiuntaochsgytieo(aaunnonpoapttohlionneatitnrhsemterh3asuia.2enlsnctpitdinphai,olce√entehre2pawex)to.htilioaeenAntrttesaneoil3orbfepnyxtalwta(oelanoinndeesc,ejiosowdaiunni3enmcsidhtonmerf·gunat·a·)dhc·sy·.citea·oigstrontrcnfaelowserr.temtewraaeiocytfhheiirsn1svtceoagolmltviehedentanrse,qaouFlfaidirsaaticdeegxrteeenstiwoon.exIten sion of Fi (formally [Fi : Fi ] = 2). By multiplicativity performed in sequence. For exthenamSpul pefurtherp, oifswe eF3 t aitoskea Qfiae(dwldjeo,imE,n i√ana3).ygfifi anrSupposenistyteeienrxrlatetring oFesni otaisonl aQsoq f(field,u√iat2rae)n ardEon oLdat tafinite(hn3fieontift utehretahxtteeor√nftsoa[iFoQpn:e(or√Qfe2]Ec,=t√. s[T3qF)hu.ea:nreF[L0e].g:=.F4[]F)=w:[oLFul:d−E1l]e][aEd:[tFo1].a: TFqhu0]ias=dra2ti.c extension. llienneg.thAncdorirnedspeeodn,d[iCng:tRIoIff]Qe=Qa(c(√2h√2wo2)f)hwliewetresnergrectetohchoc2eoormndgnsieastnntyrarueubtrcecaetsttiibcoabolrnlense,ds,ottfhrhutiiisehcsntewweuocdhoosu.yuemlIlddwftphiwelimemexcpcapholnnlyaoyunvr[modeQ[Qtibt(n(coe√a√ortcsn2e2o)sa)ntar:rs:xeuQteQrc1s]ut]ao=cat=fotncd3oa2o3en.biipgist.tsroaeaOiainednpnstpioctwoitwhnwhieeoetrnchdrooeanotnfsnhfrci2drael2eurtwdwidw≤ohhenhpaianicohch≤lidahnpv,itiselt.nsahcsnceloelle,avanrwerldlyeytmfhfaaalelsyseDe.s.etTlaTiahrhtniisswpyyiritoiehelblddlsseesmgm. eItntasllobf oils down to the fact 3 is not a power of 2! That demtuornstoruatews ethceaSnidmecigol(rna3ers)eltyrAa,uccadttdsjQiionn(ign√aian2npg)riaeansdntiaecriltrlsa3renbacutltemiiowbneaprlysocuoiunpfbtotsehnoerfmofotuwtrlmtowipcoaliue+rlcdeblxe√lste.e2andassiotaona3a(nadcdujbobviincairneyxgo)t3.evnersito h3ne, roartioonn·ea· ·losf. Sdiemgrileaerl3y., For example, Suppose F is a field, E a finitAeextensiontexttheins spiooni nofot,f itwit andeahnad vLLe ap afiniterofivniidt eextensiondexetneonusgiohn ofboaOf cE.nEk cgThen.eroTwuhene edn[Lst o[:aL nFb :lfi]i Fse=hl d][Le=d tht[ehLoer:nyEetc]o[eEspsra:orFvy]e.b√aTc2hkigissronuontdcmonastterr√uiactl,ibtlheesroelavdinegr may be surprised with the ease this nresult realCnouumldbewres caorensotfruincfitaanpcicotooeinnntdttrseragaodrduielictectsntiioiogdovntenehr3atanhctnohdedrerwwaereteseipahohlonasanva–vdeleiwpnssolgeoallvnctIveoafedn?QdenatT(tohc√htheea2fiDot)Dnfediwesliliteiasiarmnenficpopncporirtorsnoedsbsbiitnlbnleruelmauemtc.eut.bsisIbeIitantrlienagsao,ldlfowt1lbht/gnbhh3eiolysoeniilnewlswtsrhoudeadeu2stoco/eolwsda3wrttnsrnhianmaetiotosgpotchaltoycthbohoeor[neQdvfsegfia(tna.ecr√cautItat2nc3el)t3otain:hisxseQene.2n[]so.Fo=ttto:aa3Qoppib]sootw=aweienp[rFrootwohffe:2r2F!d!o0eTf]Tsh=i2hraeawt[dtFhpico:hiFnits.−c1le]arly[Ffa1l:seF.0T] =his2y.ields demonstrates the degree acts in a padrejodiinctinagbleanwyaQyiCr(ruo√ampt2oipo)nunitsamilntuhsglqetitu3sph.aeleretTe3io3enhfrxtoeateoelrlSntsenso(ciunlotmnouiottbn(ieatohrpdsnaojotoinifnottifhnoaegf)ftp.owerormfelicant+esbqi2unavroel+veec.sg2.th4e)wswoholeurtledioanl,ebao,dfc tatworeoarsaiqmtuiouanldtaralanst.eicoSueesvxnetlrienanelsaaiordnjeo.qiunaintgiosnmsnawyhbiceh the :D E][Eelia :n Fiis ps].wr wThisohbhylyew mwdemonstratese.eccaannCnnoouttlcdcoownnse stthetrcruoucn ctdegreestt√r√u22c.s.tali pcpactsopinent€dtr s inaodo uaiuct tpredictablet.sioiFdnuerattnhhdeerwrmaet oihroean,vaetlhpseolOalovnt3eechd?eertwThteehwaeDosttecialsoibanilnmissthpreurodocsbsttilhboenelmen.puerIscotienbsaglslelamorbnysolyibwlsatechdkmeogwsertnorutaitnoiogdnhtemthd3ee,adtfnaegarcemitaale3l,loytinsheten.roir·ste·ea·cadtiepnrogwmaeanryoabfne2g!slueTraphnraditsed with the ease this result rHesotwoefvetrh,iswaerhtiacvlee, nwoet wuisleldustehefincoitme peaxstsenssoiofnasr,. tIhtaits itsimexetetnosfiornms aolflyfinaistseesdeagdrdeien.g n3ew points to a chart. This Similarly, adjopinerinfogrmanedirirnatsieoqnuaelnccue2b. eForrooetxawmopuled, liefawdettoakaeIcfQu3Qbw(ic√e em2x)atwyenefisrrieostnco,enoslratrougnceettiobflQed,(et√ghri2es)ewa3no.udlFdthoiremnepxfluayrmt[Qhpel(er√,t2o)Q: (Q√]2=, √33i)s. a power of 2 which is clearly false. This yields 3 can-pblaendeone by the elementa3r.yTohpeerSatoiolnustIfiao n3fie(ld2a) lwerelows (constructible,addition, sub tthisract iwouldon, m uimplyltipli cation and di vision) mayFAioetcldctsuhriasollpnoowlyinIwaytufth,nFedwe iaeuponfrchattiahofiveneOe Olmultiplefdon€pn,lfcr lceboaevwywdwiHjidtenoeheoige denwextensionsscitFeentiavrangbcebolurliue,ismlsghewhhsmeetdadbe whn(adjoining).tatahecthvskeme.esgnnre(neToapeocunchretenosiessFquvsdsaru iswadaeora,rhyedtnrytidyiohbntfibnahiwgenaatceletcldktiekchsgcnrgotearsuahrmoenlomceculnpuiotnrbp1oainrc/odteoysdl3nrisecmnsts,mosotaondsatprsot(efet2reaxrr/orri,ua3viac.eyaccelsv)l,tIuso,te√tlsrl√tirhvu)pihy2.sec2pedh.itcresrieodmbteanhaydoveoadeutaetrntnrocb.imaoemoflnFoatoathsuyrgytmrofirbxtbuaeheelcasledlstnyrsaiumdbufrraolgrpoyesrpursrsmieamiso,sreeleatdvandnheidtwnyelswdeg.iimoitnthtAhgehntbetnhtsrhsetewrtoewaefacaposFtsoeec.aitonlthngthisesisbtstrrorueaescasutiusilcolthtsnuarcpthr.oabTlhpemioswsewrfeulmteonotlioinnemd,odnearmnely trisecting an angle and Q(√2) is the sSewtuipotpfhoasluletnFtuamkisbinaegrfisseoqldfu,tahrEe afroofiromntsitaeo+rexba2tneyntshi+oinncg2offaintcwaiehnrced.orenLOtarna,abdfit,hincceitatieoretnehrxaeatrnteidnhosanwinaoedlnsh,.oatSfvheEev.seloarTaltvtlheaedrndtj[wohLieon:iDcnFogen]lsis=amtnr[auLpycrt:boioEbenl]es[mEm.:aIFyt ]ai.nllTvbohloivsiels down to the fact 3 is not a power of 2! That Can we construct alml paoyTinohtcescuoSrnooltnuhletyiQounn-pdlearcmetha(etthhfeoemllroaawttiiiconsng,aoclnipreclaushnmoes)ut?ladnW=nc eoe3st ish(bap aevr oepowertvadkidiesencdu bofsiysnet sd2eur whichraspe3crsietsigeom nwisse inclearlytdthoot3ofhceacsnu efalse.yrr)er.asuti loThistsn.aIl nyieldsthis paa contrarticula-r case we used the dry and retahseoDnse.liHanowperovberle, mit.iAsssltaliiptpvphpeeiersddyporoueiunst3t.tt..r,iFcFwutueirvrthtehhaeoevrnrmemepootroreoeo,v,.itdthFheeoderooetenthxhoeaOeurmrgnthptwcwleoebo,waccteohkoneegnsrQstsotarqruubunclacitdsrtihioeoennrdopfipotrhetroloebdosblqnfeltue2mhmcaeeshrosiarnswwsyagereftyatmohmbepeaonQrncutoktiisvrio√gloceynrlnoe√be,dude2,nae,drninesamapmrnmreaoeostetlovelyryl√ecvidotaetnrldri,sistsebrehcyucettciainrtnenigbgaaldaQleaonengsr√oamalunvnsagigny√lalegerbageanunsdmduerpntrsis.edAwbsithratchteaelgaesbertahiiss rseuscuhlta powerful tool in modern (1) Adding an inpterfsoercmtioedn ipnosidnetqemuoefonntcwseto.raFlit2oneresset.xhaemdpegler,eief awcetstainkea prweedimctaayblfierswistaewynhulayprowgneetmcoaunlnt(ioptl2ec)oeanxnstetdrnutshcioten (f2ua.drtjhoienrintog). ( 2, 3). length mayssqqbuueaarcrioinnnggsttrhhueectc√ceiirdrc.clleIe,f,awarereehrraeevssoeollvtvoeeddcasolbibnpsyystptraeranducnactaoltlouoaagtgno.poguuo3Fssiaunagratretrgihgnoueufmrtmmfiheenonltrdrtse√adictions.,tt.hmitAoehAanobetbarshyslot etrtpandrmtahlaocaeactnrat einweatca,wslslgw,go eeohavebbrcnrmoreaanassipy histsolvedsrrosasuutsuccaultctdrcihicthona wnalotheaitqptpuhbroo eewDelianswsbetlegiareorfmkfunuelesnlinnt wtproblem.boosceyoooollmsnfiuinsenrtnprmtumriico sotItdendiev weallrderni,ntgh neboilsoatmheestl eyryrt.ersiuseltcst.inIgn athnisanpgalretiacunldar case we used the dry and At this poitnot,be(wI2fei)rFrhAaaitfvdioedanipnfiargelo.ldvaA,indtdbehijyndoeStituenDhrpniesnpoelgFuiocagst(-nieahpodlpFnabdrnapoiinecsbokgilwagne)mretfi(3.o1eom.u)2l fdnTheetaA,daonEdloiQdnnFa ie,nSolutionfiwfiag,enetladihdotnaebtathitenaicexstiotenrraercy nlsallest.pcniotoeinwiponnrotofifspvei(elotdxia√n,dnyt2ed)onifsLosutncweahdoofittQnlhciin(aotetnss2be.t)xor.utehOPcntlnxisebciaoalesenewdsoneofyolevEtsaietn.ratgeTbhelehilsenhmneewed[Lntfith:seeFlodfn]ieF=sc.e[sLsa:rEy ]b[Eack: gFr]o.uTndhims aterial, the reader may be surprised with the ease this result nCotanallwreatcioonnlseatnlrsgutwchtitcahollromrpmneoaseapitntheohtxensetmdmroianatgetiticlcehtssom,e,(oeoQ2nann)e-ctephAs√lshaodh2cofdoe.uuiltn(IldsdtnghncnaoeonorotrdtraibednbtreitenosettnqartoasatukepkWlaecsernptneiailnsobanhebgnyndray2evtvpst)hesueoh?uertiedrnphnWpcetreirmirueoiscsesoftlehenrwaa,uwstvitclhaitirethner€auheedtrct etiehroashdedowneocnoessuradosfedsbelasivarsneretfiedac rsdtosietyuaruaelcblcd ltityltathemsesw.exl..pageeInomIs nnfactgrantettuleotaonheahngtidog isot3sbeoutp tfpisosraafe iramnotnrtfirtiydaegictclukhr idueanlmla taspowerrdteihruohnecerncestaoeaissrh.sleyeids Awtoftwopeber osau2!ituicnsnr saeasThatteldw.cddtetetrhahv elaiseglde podwhybrrpryraymacaaten iinwescddtasluo canqfchume-asttiphooenwmieanrtficucolsnt–sototrhul ecintiinvmteeogrdpeeolarmynetry. the Delian problem(3.) Adding an indtIefmrFsoecnitfsitoarnafitepelosdi,tnhbtesyodtfehgtewrFeoe-cpailcratcnsleisn.wae mpreeadnicFtab,lethwaatyisuaplol npominutlstsil(pixplTe,pyehe)dxestoueSuncohts.liotuhnFtau(itroatbdnhojeotrihmnixonrgae)n,.dthye aortehelremtweontcsoonfstFru. ction problems we mentioned, namely trisecting an angle and Could awabebsstctroranacsctt2rlulaancntggupuaoagigneetosoffofiufietelsldiddtethhetehooreryaymrmattaototiohanannegnamsslwswaepetelrimarcansai,nep?opgrnlrayTaecchtsutiahicnctoarauleilslqadqiutmuneedspo3stto.ibiosbornsWneaibnietinlcanehkcucaeoovsnni.nensbstgdtyUr3reuoupsmcnucotoltrniyivpnverstseithesgrcgeaeoeotwsnoemtmdirteaheatitgrhtryvhohy.etu.ersgeyehditgrm,esptuaholeotrstn.adeni.Istntittnrhceitnsidopnaisnrtaticmhueolanhrgiscttaohsreicwavlaerduieosvueedslobtphrmaenedcnrhtyeosafnomdf athematics – the interplay If F if a fiealdle,dnibgnytgh,thmseuabFyt-rbpaelcatcnionenCgws,atemnrumuwclteteaipdcnl.oyFnIifnstgw,ruteahcnhatda(ta3vidl)eslivtpaAoilodldicinpdnotiognsisngaottrnasuen(ctwxhtin,eaytlleQ)p-rdos-epueifinclatnhcieioetndnh(taotphthnoeebienrorlatetastmhtiiooexnfnatatawlsnlpodoplnalcyainrnnottaecher,sl)eqw?s uenconstruct.ealeWemrwmienaefigyhneatslsvdhtae,eor fhdtceFi2wsrnc..icultehs,swesaedrgetamarkesensgotasmlvlolefdntboyf aannaylorgaotiuosnalrguments. Abstract algebra is such a powerful tool in modern Computing tHhoew ienvters,ewcteAtioWhWn athisevepehohani anpoint,vovteteodudfesemte mwdweoontnh slhavesitentrreacastotem eidnprovideddpvaaoslvvseerssrmayoybta ifhsimenoughatmterhrp.aepsomocIrtlrtuatalitasni ncobackgroundttstgnitmutaroraereefgnenteddoweoniofintnoifisr√titme hmhleydaeulhlltmhtyaihaismasetnatnoseor-sonrmyieuicgscatsaoladsllediadne.denmevisvanWeiwergnlloegoepnrlqpaymeaumrwaeupentnprintoartooenocitolntsfafittctmwsmreauhdatlalotiqltchybhuaeremcacmsonhtaniacatonhtir3nicetcs.cssi.tn–Ti–ncUhtgtohihpsneseosuintinbrntujstecerctcrptiposvlnla,eaydybguettohmosuiemgthrpytl.,ytuhneitdinisgtinwchtiaotnshaomulodng the various branches of Can we constrpuloecsntsgiabtlhllepccoorimnretbsipnooanntidtoleihnnesggQwtthoi-tpmhelaatchyheebs(oetfChocietoposmnreacrspaotAuirtoouitronicdnntatihgselnidaipstn.thlpceaeIlsonfuieawdn)ntie?n,edhgrwWstarehvaeceehtnhiatoaovunvesaceepolpodntrhiiosunstevmctriuucdbosocefesotdertrsdTawdeanihopanoeoaldsuitengSegtehomsa2ilnbx.ei unaenTttsvcmthkhoiteoagliovsftrnheoafiaseutbnemintloydahdnaiertanaoilasctntotpsihou,lfiluenaroetanndlildleoel,ynstswiheroexofdotumterlwpnydaodytnisonsotitpmh.arberuotelvtwteaain√kteheo2nusiebssgylmninsoeuetnratcprsoreniosqsfeutrwauticitothinbtslheweshoeilcvrheinsuglts. In this particular case we used the dry and can be donembayyontohc fieldecueralaeom mtheorynoloeynngtguansrsedye eetoemromp itproveinhengreglalyftyoioul uln3nonswrreie alnislaag tfinotteecdldidr cbconstructiblebuarramlalhnonsacwtcvhahsWeenes(cbse.a.eedsheUdaUn(vpipteo riosolvingondn3nvee.i,smdsienesecocudonotb nhnsitdnterdrtaaetficthrtrthseosieodtucungtmapg,ihlohavtmnt,tec,shuretye.tldhthmoeiTepmaoeldtidprccimiscoastsurtistinraon)scrn.cuteticioahoetnnnrsedatsnisardaemamilvliyognoinesmnitbgoghraneatnt)hih-cemeisgvatevadooaremri.oicoeauutWlsrsdyeb,ebrvaraearlnglenocechpnbhemoresatseionctofrtftuollfpyomlcoaogtnhyneeamcntaditniagcslgs–eubbtrhjaeeicctinsn,tuebmrupbtlaeysrimply uniting what should length may beractoinoCsntaorluslcdatenwdde. iIcsfoownfseltderhneugacvrtteheptcotowirncrotoesns(pnsotcouratnuntsdcthidihtbeneaegDt2pthdeoileoniinaenr2nae√tactipb2hino)ryn.otfbhatAhleietnepmsrlaeac.txlnoietoemen?rndeasTnilintophanlatrateyonsifesoa,dpniwemedgerpratemotheiseaostnyniwbsusloetsaaeiusrfi tschiewnaledbgliltcsehotaodrnloasllaroeycOncedwgtqitmsnulhaae(ntdnaeg srtdwetausadrtaaxoi gctifeestablishedgiesoehnoxttof,tefisondeubgslbitdeoatnraitnahl.o cethetnItohtieroe.yn necessaryd,teomsiaurneltdsiwppeloirc ibackgroundanatip.ornacatnicdald qmaiuvies-sitoino)n in constructive geometry. without taking sthequa Delianremmroaaottht hsproblem.eemomraatatincicsysta harirenegeenfntatinrireceliyleyr.mmOaann--mtmahmaaedodeorne.yt.ghWehWsreaeevhmeaarinarnedegc,nlneyotnohttuertntarlrulauetlplltaylleyta2reccdheotownanbivnonreeaecctnbcothtcineenihensgnetpgrsaus.oslucnaubtUtebiejopjeienncocstnftstsm,ht,sheba3becyufiuotmrntinssdotsvimidomptelhplvrpaolnecyluyegm.uhunatnTit,tiehtitrniehnmmgegsawdwtsihuchsactiaatihtnnsca.shthsiooTuanuhlsgldedeabtmwraoeinctggieettohhme cevetanrrytiu,oruaylsghebbarsaranmicchateroskpeoodflogy and algebraic number length corresptouHnrodnwisneogvuettro,weweaechahnao(vcf1oe)intnsAsotcCdrtoudouoAciurnsttldegdQditnhaw(atin√ehste2ipcnsw)ooctanioeitnsrmhtsdao,repluIctwlafthcsnietFesoutnnhpamsikaofupbvisanfoeeagtirfinrspst.etshrolqodeIoufuvt,ftcaishibdtorsiweydeotdroeftirodmhoetlriihenomenteoaFestutasroe-.goa+phrtfaloibaxoarb√nenmase2aycatlkwalohplgseyirlonaombagnuastensafa?aedinsnTobcFatinhhdveterial,aredfi,.trietdynilhOsegdoasnWivitnt m€retheheiterp swedhtoheah prspalreaderyeslovoitpirbeitnhnaoltdeti.rsinepoumt rhtnsmayoiaovn(nlaxegsds.,t c√,oyberShn)ta2iahlmt syresurprisediutsid.ltclahnahaTreotltyhtvshe,eitcrsarotyatn wibwithgsimothrttpuchoec thexdntritgsbatelnr eueaseatdcsltotyrnileovae nnithis.rndsegmien lresulteatmyheeinhtv sioslotvoferFic.al development of mathematics – the interplay hhaavveebbeeeenn oonneeiinn tthheefifirrssttppllaacceeam..papTTtlehiecrermamtsiasotsnsiucsucscohhaf araaeslseeabanlrlgtageiierbcberltrayeaicicmchtghnageineqoo-umrmmyeesaethdtrtraeyoyv.,,aealaWmlalggeoecbsbeatrnraaetilriclcanotlotottophpplteoaorlculoeoplglgayiyrntacsatnonhonddefnameaplclgatgeltieanbhbtgreeramasioicuacftbnintjcuehusmce,mtbesbmv,eeerobrnduetthrnseimmpoalstytheoumbnsiatctiuincrgieawnon.heasTtshsuehcohtuwaledsntieth century has marked Could we caodmnjoastyinruionccgtcuparonioynntilsryr(oa2uut)HnitodsoAtniewdhdareldtvtiDhsenhqeregeu,lfiaaworaranlneletoiihpwornoarntiovneaCbretglslapen(cnnmcloiartotwc.nitouuentm?shcepsoaTdtontahitsnoahttcrfteueoaiscsfot(piampmaerlrppilofnaevopescsiosdtsaiiesnbsdoqtdlesuifnaoutrnsce.eiirnrtIsecghte.lgeceoit.s.nQio4tly-n)ipmswtlaehdoceotueoslod(tslippedcftrochalreueimmgarhd)roa.atnltl geyioout.odsnageaseesql meaFurthermore,upsliaolnadngndrelaed.y)ti?niucgnWernxeetlwae hnthetaepsvdiooe inbothernd.rtiassnctuco hstwosaeesdc.h aaconstructionUrspte.ognTmhseeinsctoonfd aprobtnhyour-agthito,nathl e distinctions among the various branches of (3) Adtthdheienoogrryyanhhaiavnveteeraasecceetninottrnraalploppilnlaatccseeoifinntwchtthoaohnevecseptiprbacualellaceatsntie.ve2oeoonfgfeettohihnmeeetmmthroeyod.dfieerrrsnnt ampmplaapatclthiehce.eammtTiaoaetntriicmsciiaosanfns.au.lceThTbhrhaeaesicttawwtlegecnenhbttinrieaeitqtihchuecgcseeentonotmtuuaerrlytmyryhoh,asaatsslagmmlelbaaorrtkakheieceddrtpoparotlsogoyf manadthaelmgeabtricasi,cenvuemn btheer most obscure ones such as However, we hSaimveiln(a1ro)lty,uAasdeddjiontinhg ienacngmoiamnyntpeoaiIrIfrcfsrcse aFFucts rtifioiiofo nafna afield,lrlfipy€.eco ulIuiedtnbn, tdibyegsbeotyrthf oithetmtomhwteeao wFytfFlo oi-planebnl-ulfpeoollsdwarc.mnoilnnea gaswelwdtlcyreiut rmeanaocmstuseamedascn.sut aIaFbdfnidwcc,i enthatetsgxh(tanpetvenr eisiwossv italloaoipndlolc,e iodpnonotrlemsisinntomtrttneousarecst (atweehoxceacf,tmhy idpmentioned,oa)aeonrgtstisiunr.cetcdsTehoianh3torih.etscahcFetuon rnamelybrt)aoi.reteixholaynxmam alptrisectingnaplednl,a-ynmea,rdweee. eanlmeWm aangleeyenasttrsae roandtnf owFt i.squaringtthruslelgymceo nntnseocfting subjects, but simply uniting what should Q √3 aapppplliiccaattiioonnssooffaalleebbrraaiicctteecc1hh/n3niiqquutehes2seQ/to3toroyaalhlmmaovosesttaallcleoontththreearrlppaalrartctscseooinfnfsmmttrahuatetchhtepiemvmaelaagttieeicocssmo,,feeevtverheynen. tmthheoedmmeoronssttmoobabstschcueurmreeaootninceeisasnssu.ucchThahases twentieth century has marked may occur only(un2d()2e)ristAthhdeedCfsioonelmgtlopointsopwafuniantilgCnl t (angc(x,e1niur)tmsch weuy)Aebcemted iisuchrcnsdosotniaenornsgfpstc teorthataelhciuesntecnit(toigfpno ntarbothtrhfolemplavrcospioaldeioinr cne+xrittdnei btosandia2spnfnotodpenwn ryod+oasit ienarenhclctigei2tnroice tofelementsolsne-ts.ipewnaldovahcoochelelirovneoc(eeftacss hiu,.oftebrsh,) er.cF.aosa toCanoirorleudnrtiana ilotaniptotheelnaoshanfale tsnvcircle,)w.ed?oSbteWshevieme nrareuhauolatnsva eednresolvedtjediohnoieusnctsichunolesgiosnsr efiedbymraisrnaat yaeanalogouspstqbeluageacmxeti.eonsnTttseoorw fmoarguments.habsintcayshiunrcahtthiaoesnd aaAbstractellsgierebdrapico igneto. metry, algebraic topology and algebraic number performed in sequencle.nccogFontonhsrsttmreruxuacacytmtivibvpeeelegcg,eoeionfomsmwtereetutrtcryayt.ke.ed.QIfwweemahapaypvelfiicrtsaotticeoonnnlsastrorgfueactltoebaQrpa(oi√cin2tt)ecianhnntdihqteuheresanttifounraatlhml eporlsatnoaeQl,lw(o√eth2me,r√ayp3a)sr.ttasrtofwmitahthseegmmaetnictss,oefven the most obscure ones such as (1) Adding an(i3n)teArsdecdctaiinnognwebapen oconstructdinot(ne2oer)fsebtAcwytdoidt oallhnilnieng ppointseeosali.enmtisCenn tooonetfuartls rdwtheeycowtoiec opinreccro-placeplanetosiit.onrntusc o(thetfapafio leirationalinlndtesaolnulodtws aplane)?isdce(iartcdhldei. triaotalgebranio,tnhsaeulobprtyl raisanh csucheat?ivoenT, ahama cpowerfulteunilstiripamllipcpaolt satoolisocienbliean innud tsmodernhinedgivpoiasnlialoy tnmathematics,et)hoef stthreaigmhotde eonedrgnem alaotnhee.matician. The twentieth century has marked Suppose F is a field,lEengatfihncitoerreexstpeonnsdioingotfoiteanchd oLf aitcsfioncnoistoterduexicntaeivntessigoaenonmodfetEthre.yn.Tuhseent[hLe :cFoo]r=din[Lat:eEa]x[Ees :toF ]o.bTtahiins the desired point. (2) AddingCaonmipnutetrinsegcwtihotehnWeoipnutot ehaveinrtsat(ek3oci) tnfdiscussedigoaAndsliqdnpuieonaiganrentaH darono fasegmentowitntcwetsiveroorecsrllei,n.acwetn iseofyotnihnh anyavpivnooeglivn nrationalfetaossnttochufiestrew.sd oolengthOltuhcntierioctcnhloee smmayo.ofpttahws ebesorssh oiamfnaudrla,.tpatIpnhtleiiocslauattstiimtoleinnersetaoworfofaeolqcreoumbnarastaltiilorcyuntcsaetscwisohehnsisicqahmdudeasiyntgoinnavelomwlvoepstoianltlsottohear cphaarrtts.oTf hmisathematics, even the most obscure ones such as demonstrates the degreeCaocutlsdinwea cpornedstircutacbt lpeowinatys uopuotsnidme utlhteiprlaeteioxnteanl spiolanne(a?dTjshouldohinaitnigs) .inotmpo bessi btakenle usi nbyg o surprisenly the swithtraig htheset edg eresults.alone. In this (3) Addingcan ibnetedrsoencetiobnyconstructed.ptohientesleomf etnw tIfoa rcweyirco mlhavepease.yraot toicocnu constructsr aonfilyeludnad lael ropointwthse(af oindldl otheiwtioin nrationalg, csiurbcutrma cstaioncnco,ensmst(urpulrtcoitpvilividceeadgteiioonnmteeartnsredyc.tdioivnissidoon)occur). CoHmopwuetvinerg, twhee hinatveersneocttiuosnedpotihnet coofmtwpoaslsinseosfianrv.oIltveis thime esotparticularloutfioornmoafll ytcasewaossse ismweaudl tdusedainego nutheeswli npdryeoairn tandesqtuoa abstracttaiocnhsawrth. ilanguagecThhis of Computing thewiinttheorusetctaiokninpgoplane,sinqctuaanorf ewebtweroo domaytolsinneoe rsstartbaiynnvyto htwithlhev(ien1esg)l etsegmentsmhfAaeednndsctoiieanlrug.ytiaoOo nofnpnieon trlengthfthaeterwisooeontcshst icorrespondingeomarnuhfilpaetolandindne,taoltoulhofsewtlwslian ot(etaalefielddirnrdetisqtw.i uotheoryantc,ioosnnussbtr wtutorhacict cansweritohionns,mma uyal tpracticaliinpvloiclavteion questionand div iinsi oconstructiven) may occur only un(d2e)rAthded3ifn.oglTloahwneininSgtoeclrirsucetucitmoiosnntapnocienst(opfroavliidneedaindtearseccirtciolen.s do occur). can be done by the elementatory weachoitpheor auoftio titsnask icoordinatesnagfiseqlduaarlelorw oandsot(s athenodrdiatni ouseynt,h isntheugbtf arcoordinateanctieiorn. ,Omnu axestlthieplo ictahgeometry.teironhaanndd, td hiveislaiotnte)r two constructions may involve (1) Adding an(3in)teArdsedcitnigonanpoinintterosfectwtionlipneosin. ts of two ci3rcles. without taking sqAutatrheirsopootsinotto,r wa obtainneyhtahvine gtheprfao nvdesiredcidieerd. eOn point.onutghhe boatchkegrrhouand,otnhefielaldtttehretowryo tcoonpsrtorvuect√io2nsismnaoyt icnovnosltvreuctible solving (2) AddingCoamn pinutteinrsgectthieoninptoerinsetcotfioanlpinoeinatnodf atwcoircllien.es involves the solution of two simultaneous linear equations which the Delian problem. We have demonstrated a very important trend in (3) Addingcaannbinetedro2sneectiboyn tphoeinetlsemofetnwtaorycirocpleesr.ations a field allows (addition, subtraction, multiplication and division) If F if a field , by the CouldF -plan e wewe constructmean F , t pointshat is al outsidel points ( thex, y ) rationalsuch th at thebot hhistoricalx and y adevelopmentre elements o fofF .mathematics – the interplay Can we construcplane?t allCpo omThatinptust oiisnn gimpossibletthheweQiitn-hptoelaur sctusingeectta(itkohi nneonlygpraostqi nitheuotanorae flstraighttprwolaontslei)no? eredgesWainneyv thoalone.hlavivneegs dtf ahisneccusisoesrleu.dtOioansetoghfmetweonothtseoirmf huaanltnyadnr,eatothiuoesnllaailntteearr tewqouactoionnstsruwchtiicohns may involve can be done by the elementary operations a field allows among(additi oseeminglyn, subtra unrelatedction, mu lbranches.tiplication Uponand seconddivision thought,) length may be coHowever,nstructed we. If havewe h anotve tusedo con thestru compassct a poin tsoin far.th eItr aist itimeonal topla ne, we may start with segments of length correspondingwtiotheoaucth toafkiitnsgcosqourdairneartoeostasnodr tahneynthuisnegthfaencoieorr.diOnantethaexeothesthteo rdistinctionsohbatnadin, the dlaamongetstierredtw ptheoicno tvarious.nstruct ibranchesons may iofnv omathematlve - formally asses adding new points to a chart. This may occur ics are entirely man-made. We are not trully connecting sub Could we cononlystru cundert poin tthes o ufollowingtside the rcircumstancesational plane? (providedThat is im intersecpossible- using only the straight edge alone. However, we have not used the compass so far. It is time to formally asses addjects,ing ne butw p osimplyints to unitinga char twhat. Th ishoulds have been one in the tions do occur). may occur only under the following circumstances (provided intersections do ocfirstcur) .place. Terms such as algebraic geometry, algebraic to- pology and algebraic number theory have a central place (1) Adding an in(1)te Addingrsection anpo intersectionint of two li npointes. of two lines. (2) Adding an intersection point of a line and a circle. in the palate of the modern mathematician. The twentieth (3) Adding an in(2)te rAddingsection anpo intersectionints of two c pointircles .of a line and a circle. century has marked applications of alebraic techniques to (3) Adding an intersection points of two circles. Computing the intersection point of two lines involves the solution of two simualmostltaneou sallli notherear e qpartsuatio nofs wmathematics,hich even the most ob- can be done by the elementary operations a field allows (addition, subtractioscuren, m uonesltipl isuchcatio nasa constructivend division) geometry. without taking s quare roComputingots or anyt h theing intersectionfancier. On pointthe o t ofhe r twohan d lines, the inla-tter two constructions may involve

Spring 2009 29 Mirror Neurons

Elizabeth L. Robinson

ntertainment, in spite of its great variety across portant because it is the mechanism by which Etime and culture, is really just about one we make sense of other’s actions and intentions. thing: watching. From movies to sports to tra- Without this ability, social behaviors including mirror peze artists, there is something about watching communication, cooperation, and imitation learn- other people that we just can’t get enough of. ing would be much more difficult. But shouldn’t we prefer doing to watching? The Although they are called “mirror neurons,” study of a new class of brain cells, called “mirror the cells themselves do not actually represent ac- neurons,” may offer some insight. Although these tions; they remember and recreate patterns. You cells were discovered in 1996, it is only now that can, for example, imagine your brain as a grid of we are beginning to understand their significance. thousands of tiny light bulbs: the lights are mean- It was a day like any other in Giacomo Riz- ingless when lit individually, but certain groups of zolati’s neuroscience lab in Parma, Italy. Rizzolati them turned on at the same time can make let- and his team were busy collecting data for a study ters, pictures, and patterns. These patterns, rather on the brain areas that control hand movements than the individual brain cells, represent informa- in monkeys. They had implanted wires in certain tion. A mirror cell functions by activating the brain cells so that whenever the monkey reached specific group of neurons that creates the action for a peanut, the brain cells involved would make when you perform it. In fact, the only reason you

a buzzing sound signaling that they were active. don’t literally “ape” the person you are watching The study was proceeding as expected until, one is that cells in your spinal cord stop these motor day, there was a buzz from a monkey sitting per- signals from reaching your limbs. fectly still when a researcher reached for a pea- From studies using functional magnetic nut. Somehow, the monkey’s motor neurons were resonance imaging (fMRI), an imaging technique responding as if the monkey was doing exactly that records brain activity by monitoring blood what it saw the researcher doing. This finding flow, Oxford neuroscientist J. Grezes and others was so unexpected that, initially, the researchers have shown that mirror neurons are also present suspected their equipment was faulty. However, in humans. Although fMRI is less precise than their observations were confirmed repeatedly in direct-cell recording (as was used in the primate a number of different contexts. The scientists research referenced earlier), there is evidence that were baffled and, according to Rizzolatti, “it took human neurons are actually far more advanced several years for us to believe what we were than those found in other species. Our mirror

mirror seeing”(Blakeslee). He rather poetically named cells respond to the individual sub-movements the cells “mirror neurons” and continues to study that make up an action rather than only to the ac- them today. tion as a whole. Also, our mirror cells respond to Initially (and more aptly) known as “mon- people’s movements regardless of whether or not key-see monkey-do neurons,” these cells allow they have a definite purpose, unlike monkey mirror us to understand what other people are doing neurons, which only respond to directed actions. by, essentially, mirroring them. Our minds actu- This seemingly trivial difference is significant from ally act out what we see others doing. Although an evolutionary perspective because it allows for this capacity may seem trivial when talking about the formation of new movement sequences and, peanuts, the ability to “mirror” is extremely im- as a result, complex learning and creative innova- tion. This in turn enables individuals to recreate an action through observation and, subsequently, These cells allow us to under- to pass it on to the next generation. It is clear “ that this ability to parse actions into smaller units stand what other people are offers a huge evolutionary advantage. Imagine trading in your paper airplane for a box of legos: doing by, essentially, mirroring suddenly the possibilities are endless. them. ” According to neuroscientist V. S. Ra- 30 Columbia Science Review Mirron Neurons machandran, the existence of mirror cells may solve an abiding mystery of human evolution known as “the great leap forward” in which, 40 thousand years ago, sophisticat- ed technology, art, clothing, and culture emerged suddenly and without apparent cause. If mirror neurons began parsing information at this time, it would explain how individuals were suddenly able to learn so much faster. Interestingly, the mirror system also encompasses aspects of emotion. According to Marco Iacoboni of UCLA, the mirror system is the mechanism that underlies our abil- ity to empathize. Mirror neurons allow us to respond to another’s experiences as if they were our own: we cringe when we see someone get hurt and cheer when an athlete scores. We, quite literally, feel each other’s pain. This ability to make sense of other people’s actions also facilitates the creation of social groups by allowing individuals to under- stand others’ efforts at communication and cooperation. Beyond this, a study by Lisa Aziz-Zadel at the Uni- versity of Southern California shows that our mirror cells can also be activated by descriptions of actions, both heard and read. Based on these findings, some speculate that mir- ror cells played a crucial role in the emergence and evolu- tion of language, as well as in language learning. One could speculate that mirror cells are part of the reason why even the rowdiest of teenagers can spend hours on end glued to a Harry Potter book. They may be sitting still, but in their brains they are catching golden snitches and dueling with evil wizards. We may only be scratching the surface of the poten- tial of mirror neurons; for instance, they may have important nections. They are always influencing each other and none applications in the study of autism. Autism is characterized ever acts alone. Mirror cells are, however, distinctive in one by impaired language and social skills, both of which are, significant way: they only function in a social context. That as we have seen, associated with mirror neurons. A study is, they are part of the biological mechanism that underlies by Ramachandran and Lindsay Oberman at UCLA monitored our ability to understand other people. The fact that we the brains of ten autistic children while they watched vid- have neurons such as these proves that we are hard-wired eos of other people’s hands moving and while they watched to be together. their own hands moving. They discovered that these chil- References dren’s mirror systems were only activated by watching their own hands, suggesting that they are unable to internalize Rizzolatti, G., Craighero, L. “Mirror neuron: a neurological approach to empathy.” other people’s actions. Hopefully, further studies will lead Neurobiology of Human Values. Springer-Verlag Berlin Heidelberg 2005. to a better understanding of autism, which according to the Rizzolatti, G., Craighero, L. “The Mirror-Neuron System.” Annual Review of Neuro Center for Disease Control and Prevention currently afflicts science. 2004. 27: 169-192. Blakeslee, Sandra. “Cells that Read Minds.” The New York Times. 10 January 2006. one out of every one hundred fifty children in the United Gallese, V., Fadiga, L., Fogassi, L. and Rizzolatti, G. Brain 119, 593–609 (1996) States. http://brain.oxfordjournals.org/cgi/content/abstract/119/2/593 Since their discovery, mirror neurons have been im- Damasio, A., Meyer, K. “Behind the Looking-Glass” Nature 454, 167-168 (10 July 2008) plicated in discussions of a broad range of topics including Ramachandran, V.S. “Mirror Neurons and the Brain in the Vat.” Edge: The Third Cul sports fandom, physical therapy, altruism, and even pornog- ture http://www.edge.org/3rd_culture/ramachandran06/ramachan raphy. However, it is important to acknowledge that these dran06_index.html Swanson, Jeanene. “Mirror Neurons Also Respond to Language and Sound,” Seed ideas are still speculative and more research is needed be- Magazine. September 21, 20 fore the role of mirror cells can be completely defined. In Iacoboni, M., Woods, R., Brass, M., Bekkering, H., Mazziotta, J., Rizzolatti, G. “Cortical Mechanisms of Human Imitation.” Science 24 December neuroscience, as in many other areas of study, it is rarely 1999: Vol 289. No. 5449, pg. 2526-2528. http://www.sciencemag.org/ accurate to explain a problem as the direct result of a single cgi/content/full/286/5449/2526 cause. The human brain is made up of billions of neurons Grezes, J., Armony, J., Rowe, J., Passingham, R. “Activations related to “mirror” and “canonical” neurons in the human brain: an fMRI study.” NeuroImage that are constantly sending signals and changing their con- Volume 18, Issue 4, April 2003, pg. 928-937.

Spring 2009 31 Beans

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32 Columbia Science Review