sign in sign up
<<
Home , Ernst Kummer, Kurt Hensel

p-adic Emoonah McClerklin Overview

P-adic numbers are a complex field of prime numbers that serve as an extension of the set of rational numbers. Created by , P-adic numbers are a fascinating phenom- ena that have been used primarily in theory. P-adic numbers have been seen in famous mathematical problems throughout time from the (bloop)to the Shimura-Taniyama conjecture and Andrew Wiles’ proof of Fermat’s Last Theorem.

To the average person, the words used to define P-adic numbers might seem strange and alien. P-adic numbers cannot be understood without the understanding of some basic mathematical terms. First, rational numbers (Q) are the numbers made by dividing two intergers (a number with no fractional part). 1, 1.35, and 1/2 are all examples of rational numbers.

A field is a set of numbers that satisfy the field axioms (rules about fields that have not been proven but are generally believed to be true). There are 12 field axioms: The existence of ad- dition, the associativity of addition (a+(b+c)=(a+b)+c), the addititive identity (a+0=a for all a), the additive inverse (a+(-a)=0 for all a), the commutativity of addition (a+b=b+a), the existence of multiplication, the associativity of multiplication (a*(bc)=(ab)*c), the mul- tiplicative identity (a1=a for all a), the multiplicative inverse (aa−1 for all a6=0) The com- mutativity of multiplication (ab=ba if a nd b are real numbers), the dstributive property (a(b+c)=ab+ac), and the non-triviality axiom (06=1). So P-adic numbers are a complex set of numbers that serve as an extension of the set of numbers made by diving two intergers.

P-adic numbers serve a multitude of functions, but in this paper we will focus on p-adic intergers. For a prime (p) the ring of P-adic intergers, denoted Zp, is the inverse limit of the set of the congruence module pn. It is written as:

lim /pn = ←− Z Z Zp n

To be understood, this definition must be broken into layers. Rings are a set of numbers with two binary operators (+ and *) that satisfy five axioms: additative associativity, additative commutativity, additive identity, additive inverse, left and right distributivity (For all a,b,c∈( that is a member of) S, (a*(b+c)=(a*b)+(b*c)), and Multiplicative associativity.

The term congruence class refers to modular arithemtic. Modular Arithmetic does not follow the conventional rules regular arithmetic follows. When dividing two intergers we get the equation:

A = B · Q + R

Where A is the dividend, B is the divisor, Q is the quotient, and R is the remainder. In modular arithmetic we only care about the remainder. In modular arithmetic A ≡ R mod B

(if n|b−a) where A, B and R still represent the dividend, divisor, and remainder respectively. For example, when looking at congruence modulo 5 :

7 ≡ 2 mod 5 107 ≡ 2 mod 5 100000000007 ≡ 2 mod 5

n The set of congruence class module n is denoted Z/nZ. The notation Z/p Z means the congruence class module pn. Therefore, the p-adic interger equation lim /pn = is ←− Z Zp the inverse limit of the congruence modulo pn.

To understand inverse limits we must first understand inverse systems. An inverse system is an indexed family of groups whose index set is a directed partially ordered set,together with a set of homomorphisms (which are also indexed) such that the domain and codomain of each homomomorphism is obtained by reversing it’s ordered pair of indexes.

Like many other definitions, the meaning of inverse systems must be explained in layers. First, we must understand the term family. A family is a term used to describe a collection of objects. A family is denoted {ai}i∈I where I is a nonempty set called an index set (set whose members label members of another set) and ai is called the term of index (i).

A group, denoted G, is a set of elements, either finite or infinite, that have a binary operation (addition, multiplication, subtaction, or division) called the group operation that together adhere to the ”four fundamental properties:” closure (If and only if A and B are elements in G, then product A ∗ B is also in G), assocativity, the identity property, and the inverse property. Groups with a set amount of numbers are finite groups. The number of elements in a finite group is the group order. If there is a closed group within a group that group is called a subgroup.

An important group to understand is a cyclic group. A cyclic group is an Abelian group (a group that is commutative) that can be generated by a single element, X. This means that all of the numbers in the group can be produced by applying X in some way. Cyclic groups with a finite group order n (a fixed number of elements where n represents the fixed number) are denoted Cn, Zn,Zn, or Cn. The group Z/nZ is an example of a cyclic group. Partially ordered sets are sets that have a partial order, or, a set whose relation (ex. ≤) has reflexivity (A ≤ a for all a ∈ S), antisymmetry (a ≤ b and b ≤ a implies a = b), and transitivity (a ≤ b and b ≤ c implies a ≤ c).

A homomorphism is a map between two groups that adheres to the group’s partial order, identity, and group operation. If A and B are two groups, and f maps A to B (for every a∈A,

2 and f(a) ∈ B) Then f is a hormorphomism, where every a1, a2 ∈ A, f(a1a2) = f(a1)f(a2) because the range (output) of f adheres to the structure of Group A and B. For example in the following equation I is the group of all interngers under addition, and tge group G = [1, −1, i, −i] such that f(x) = in for all n ∈ I. Consider the equation:

f(x) = in, f(m) = im, ∀m, n ∈ I

f(m + n) = im+n = im ∗ in = f(m) ∗ f(n)

This scenario is also an example of a homomorphism because range of the mapping f adheres to the structure of the Groups I and G.

Now that we understand families, posets, indexes, index sets, and homomoprhisms we can deconstruct the definition of an inverse system. An inverse system is a collection of groups whose index set not only has a partial sequence that adheres to the rules of reflexivity, antisymmetry, and transitivity, but also is paired with a map between two groups (homo- morphism) whose subset corresponds to the sequence of the partial order, such that the domain and codomain of the homomorphism is obtained by reversing the ordered pair of indexes. In laymans terms this means that an inverse system is a bunch of sets of numbers that have structure. Numbers between one set can transform to the other yet still adhere to the structure of the first via a map between the groups that transforms the numbers with a binary operation.

Inverse systems give rise to inverse limits. An inverse limit can not exist without first having an inverse system. Most inverse systems are created with the intention of producing a inverse limit. The concept of inverse limits is to complex to discuss in depth in this paper, but we can try to form an understanding by studying a specific case: p-adic intergers.

A p-adic number is just a sum of multiple powers of p, where p represents any . The p-adic expansion Z/pnZ can be written with any rational number and produce the output 1 p−1 . For example:

Z/5Z

means

∞ X 5−n = 1 + 5 + 52 + 53.... n=1

Under normal circumstances this number should diverge, but instead it comes out to be 1 . That’s because p-adic numbers don’t follow the rules that rational follows. In the p − 1 same reasoning, p-adic intergers don’t follow the rules that normal intergers follow. In Z/pnZ one group of p-adic intergers can be mapped to a lower group using it’s reciprocals. Keep in

3 mind that p-adic intergers follow the rules of modular arithemetic, so the reciprocals of the product of prime numbers are calcualted differently than normal. Take the equation:

lim /5n = ←− Z Z Zp

This is the same as writing:

2 3 4 Z/5Z ←− Z/5 Z ←− Z/5 Z ←− Z/5 Z

Each group correlates to a different summation. For example, the same map can be written as:

1(1) ←− 1 + 5(6) ←− 1 + 5 + 52(31) ←− 1 + 5 + 52 + 53(156)

If we take these summations and write them in terms of modular arithemetic we find that the mapping of these numbers are extremely unique. For example,

2 3 4 Z/5Z Z/5 Z Z/5 Z Z/5 Z 1 ←− 6 ←− 31 ←− 156

When using modular arithmetic the remainder of one group becomes the summation of the 3 2 next. For example 156 mod5 is 31. 31 mod 5 is 6. 6 mod5 is 1. The mapping of Z/5Z is an example of an inverse limit. History of P-adic Intergers

The history of p-adic intergers can be traced back to . Ernst Kummer was a German mathematician born on January 29th, 1881 in Sorau Brandenburg (which, at the time, had been part of ). Kummer’s father died when he was only three years old, and so he grew up with his mother. As a child he recieved private schooling before entering Sorau’s Gymnasium at 9 years old. In , a gymnasium, is the most advanced form of secondary school. It is similar to a college prepatory highschool in the US. Most gymnasiasts (people admitted into a gymnasium) start school between the ages of 10 and 13, and stay there for about 9 years. Kummer was an intelligent child, being admitted to Sohou’s gymnasium at the age of 9. In 1828 Kummer attented the University of Halle. At first, he wished to study Protestant theology, but later decided to study math with the intention of using it as a foundaton for philosophy. However, as he studied his interest for mathematics grew and he switched his main line of study to mathematics.

In 1831 Kummer won a prize for his essay on topic sets. He was then awarded a doctorate for the strength of his essay. He also won a certificate that allowed him to be a teacher. During 1831 and 32 he taught a Sorau’s gymnasium before being given a teching position

4 at the gymnasium in Leignitz, which is now the city Legnica in Poland. Kummer taught mathermatics and physics in Liegnitz for 10 years, at the same time conductin mathermat- ical research. In 1836 he published a paper on hypergeometric , which he sent to the famous mathematician Jacobi. Later, Jacobi and Dirichlet, another renowned mathemati- cian, collaborated with Kummer about various mathematical topics. Jacobi and Dirichelt, realizing Kummer’s mathematical genius, set out to find Kummer a professorship. In 1839, under Dirichlet’s recommendation, Kummer was elected to the Berlin Academy of Science, a society of renowned scientists, including famous mathematicians like Lagrange and Euler. In 1840 Kummer married Dirichlet’s cousin. In 1842, with the help of Jacobi and Dirichlet, he was appointed a professor at the University of Breslau, now Worclaw, Poland. At the Univeristy of Breslau Kummer started to explore . In 1848 his wife died, but he remarried fairly soon.

In 1843 Kummer realized something crucial about Fermat’s Last Theorem. Kummer theo- rized that Fermat’s Last Theorem could not be proved because the ring of integer’s unique factorization could not be extended to certain cyclotomic fields. In the ring of intergers every number can be written in a unique way as a product of prime numbers. For example, the prime factorization of 28 is 22 ∗ 7, the prime factorization of 572 is 22 ∗ 111 ∗ 131. However in the ring of intergers of cyclotomic firleds, or cyclotomic intergers, some numbers do not have a unique factorization. Kummer attempted to reconcile this issue by exploring the idea of ’ideal’ numbers, which restores unqiure factorization for cyclotomic intergers. Kummer dis- covered that facotorizing cyclotomic integers into the product of ideal prime numbers solves the issue of cyclotomic intergers not having a unique factorization. Kummer’s theorem is used to find all of the ideal prime numbers in a given natural prime number.

Richard Dedekind, another German mathematician took Kummer’s theorem further. Dedekind’s father was a professor at Collegium Carolinum in Brunswick, Germany. He went to Martino Catharineum in Bruncswick when he was seven. At the time he was interested in physics and chemistry. However, Dedekin did not like the lack of structure in physics and turned to mathematics, which he thought ha a precise logical structure. At age 16 Dedekind entered the Collegium Carolinum, which was an institution that was between a highschool and a college. There, he furthered his understanding of mathematic. In 1850 he entered the Uni- versity of Gottingen with a solid mathematical background. The University of Gottingen was not yet the renowned university it would become. At the time Dedekind attended the school, renowned mathematicians like Stern and Gauss taught classes. The mathematics department and physics department at Gottingham combined to conduct seminars, which Dedekind regularly attended. At these seminars he learned about number theory.

Dedekind did his Doctorate work under Gauss and recieved his doctorate in 1852. He was the Gauss’ last pupil but he was not trained in advanced mathematics. The most advanced form of mathematics he new was number theory. After he recieved his doctorate Dedekind spent the next two years learning more advanced mathematics, and the newest mathematical developments. In 1854 he began teaching probability and gemoitry at Gottingen. In 1855, after Gauss’ death, Dedeking found himself working alongside Dirichlet. Dirichlet taught him theory of numbers, potential theory, and partial differential equations. They became close friends and Dedekind’s mathematical interests took a new direction after Dirichlet’s

5 guidance.

The extension of Kummer’s theorem can be attributed largely to Dedekind. According to Fernando Q. Gouvea, Dedekind generalized the method to algebraic number fields. Dedkind defined the ideals for rings of integers of an algebraic number field. An ideal is a special subset of rings that generalize certain subsets of intergers like ever numbers, multiples of 3, etc that adhere to certain closure and absorption properties. The describes an integral domain in which nonzero proepr ideals can turn into a product of prime ideals. Dedekind created this theorem based off Kummer’s work.

Dedekind also worked with Weber to further Kummer’s theorem. In 1880 they wrote a paper about the theory of algebraic functions of one variable which connected Kummer’s concept ideal numbers to one variable functions. According to Dedeking and Webber every ideal can be broken down into finite parts called prime ideals. Prime ideals correspond to linear factors in the theory of integral rational functions.Using the concept of idea; Dedekind and Webber gained a general definition of ”a point of the Riemann Surface”.

The concept of p-adic numbers also has roots in the work of . Weistrass was born in 1815 and grew up in Prussia where he moved from school to school because of his father’s job as a tax inspector. In 1829 he attended the catholic gymnasium in Paderborn. At the gymnasium Weierstrass exceled in mathematics. However, Weierstass’ father wanted him to study finance, so when he graduated from the gymnasium and attended the , that’s what he did. Weierstrass loved mathematics, but was torn between pursuing the subject and adhering to his father’s wishes. This conflict caused Weierstrass to sieze up and stop attending lectures all together. While in college he pretended not to care about his academics , and instead immerse himself in the sport of fencing and the activity of drinking.

In college Weierstrass did study mathematics on his own. He read Laplace’s Mecanique Celeste and Jacobi’s and Gudermann’s work on elliptic functions. While seceretly studying mathematics Weierstrass decided that he would study math for his career, but he still kept the appearance that he was heading on a finance track. He stayed at the University for a year after making his decision then dropped out altogether without taking any exams. Later Weierstrass’ father let him attend the Theological and Philosophical Academy of Munster so that he could become a secondary teacher. In reality, Weierstrass just wanted to attend the lectures of Gudermann, a mathematician who was working at the Academy in Munster. Gudermann became Weierstrass’ mentor and encouraged him in his pursuit of mathematics. Gudermann felt that Weierstrass was a smart as any of the renowned mathematicians with published papers.

In 1841 Weierstrass began teaching at the gymanisum in Munster. He didn’t publish any mathematics during that time but he did write three short papers between 1841 and 42. Later he taught at a prestigious gymnasium in Prussia where stayed until the he moved to the Collegium Hoseanum in Braunsberg in 1848. At the Collegium Hoseanum he taught not only math but physics, botany, geography and history among other subjects. During his time teaching Weierstrass became sick for a prolonged amount of time. He suffered from sudden bouts of dizziness that resulted in severe sickness. His health cause the mathematical

6 papers he did publish to go unnoticed. He published papers on abelian functions but got no recognition. In 1854 he published a Zur Theorie der Abelschen Functionen which became widely popular.

With his paper Weierstrass became famous in the world of mathematics. He recieved an honorary doctor’s degree from the Unviersity of Konigsberg in 1852. In 1855 he gained Jummer’s vacant chair at the Universirty of Breslau. He also became a senior lecturer at Braunsberg. Afterwards he took a break from academics for a year to work on his mathematics. Eventuall Weierstrass worked on the concepts that influenced the creation of p-adic numbers.

The person to take these concepts and create the idea of p-adic numbers was Kurt Hensel. Hensel was born in the East Prussian city formerly knows as Konigsberg. Hensel was born to a line of famous painters and musicians. He also is related to Kummer, a man whose mathematical concepts he would use to create his own. While living in Konigsberg Hensel was homeschooled until he was 9. Later his family moved to Berlin and Hensel attended the Friedrick-Wilhelm Gymnasium where he was taught mathematics by K H Schellbach. Under Schellbach’s guidance Hensel began to foster a deep love for mathematics. When Hensel left highschool he knew for sure that he wanted to go into mathematics. In coolege Hensel studied in schools in Berlin and Bonn. He was taught by famous mathematicisn like Lipschitz, Weierstrass, Borchardt, Kirchhoff, Helmhotz and Kronecker. kronecker had the greatest influence on Hensel. He over saw Hensel’s doctorate studies at the Univeristy of Berlin.

In 1887 Hensel married Getrud Hahn in Berlin in 1887. Getrud was the aunt of Kurt Hahn, the founder of the renowned Gordonstoun School in Scotland. Hensel and his wife had one son an three daughters. In 1901 Hensel received full professorship at the . He spent the rest of his life in Marburg, retiring from the university in 1930. During his time at the university Hensel spent a great amount of time aditing Kronecker’s works. Between 1895 and 1930 Hensel published five volumes of his mentor’s work. Hensel also wrote a eulogy for Kummer’s work Gedachtnissred auf Ernst Eduard Kummer.

Hensel chose to devote his individual work to the development of arithmetic in algebraic num- ber fields. In 1897 Hensel developed the idea of algebraic numbers. He combined Kummer’s concept of ideals, Dedekind’s application of ideal to one variable functions, and Weierstrass’ method of developing poer-series for algebraic functions to create p-adic numbers. Hensel was interested in finding the power of a prime that divides the discriminan, a function of coefficients that indicates when a pair of roots of a polynomial are equal, of an algebraic number fiel a subfield of R or C that contains rational numbers and polynomial roots with rational coefficients.

7 Glossary

Cyclic Group - is an Abelian group (a group that is commutative) that can be generated by a single element, X.

Index Set - A set whose members index (label) members of another set.

Inverse System - an indexed family of groups whose index set is a directed partially ordered set,together with a set of homomorphisms (which are also indexed) such that the domain and codomain of each homomomorphism is obtained by reversing it’s ordered pair of indexes

Intergers - numbers with no fractional parts

Group - a set of elements, either finite or infinite, that have a binary operation that adhere to the four fundamental properties:.

Group Operation - The binary operation of a group

Homomorphism - a map between two groups that adheres to the group’s partial order, identity, and group operation.

Family - a term used to describe a collection of objects.

Field - set of numbers that satisfy field axioms

Field Axiom - rules about fields that have not been proven but are generally believed to be true

P-adic numbers - a complex field of prime numbers that serve as an extension of the set of rational numbers

Partial Ordered Sets - Sets with a partial order

Rational Numbers - numbers made by dividing two intergers

Ring - a set of numbers with two binary operators (+ and *) that satisfy five axioms

8 References

1. http://www.math.uic.edu/\someone/notes.pdf

2. Bak, Newman: Complex Analysis. Springer 1989

3.https://www.math.purdue.edu/ rcp/FieldAxioms.pdf

4.http://mathworld.wolfram.com/Field.html

5.http://math.mit.edu/classes/18.782/LectureNotes4.pdf

6. http://www.math.mtu.edu/mathlab/COURSES/holt/dnt/cong1.html

7.https://ncatlab.org/nlab/show/p-adic+integer

8.http://www.math.wm.edu/ vinroot/307ModNote.pdf

9. http://www.math.cornell.edu/ mec/2008-2009/Victor/part6.htm

10. http://www.yourdictionary.com/inverse-system

11. http://people.virginia.edu/ mve2x/7752 Spring2010/lecture26.pdfhttp://people.virginia.edu/ mve2x/7752 - Spring2010/lecture26.pdf

12. http://www.math.wm.edu/ vinroot/PadicGroups/limits.pdf

13. http://www-groups.dcs.st-and.ac.uk/ history/Biographies/Kummer.html

14. http://www-history.mcs.st-and.ac.uk/history/Societies/Berlin.html

15. http://www-history.mcs.st-and.ac.uk/Biographies/Dedekind.html

16. https://en.wikipedia.org/wiki/Dedekind domain

17. https://www.encyclopediaofmath.org/index.php/Ideal number

18. http://the-paper-trail.org/TranslationOfDedekindWeber.pdf

19. http://math.uchicago.edu/ may/REU2014/REUPapers/Lee.pdf

20. http://www-history.mcs.st-andrews.ac.uk/Biographies/Hensel.html

21. http://www-groups.dcs.st-and.ac.uk/ history/Biographies/Weierstrass.html

9