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AMS / MAA TEXTBOOKS VOL 23 Learning Modern From Early Attempts to Prove Fermat’s Last Theorem

Al Cuoco and Joseph J. Rotman i i “book2” — 2013/5/24 — 8:18 — page i — #1 i i

10.1090/text/023

Learning Modern Algebra

From Early Attempts to Prove Fermat’s Last Theorem

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c 2013 by The Mathematical Association of America (Incorporated) Library of Congress Control Number: 2013940990 Print ISBN: 978-1-93951-201-7 Electronic ISBN: 978-1-61444-612-5 Printed in the United States of America Current Printing (last digit): 10987654321

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Learning Modern Algebra

From Early Attempts to Prove Fermat’s Last Theorem

Al Cuoco EDC, Waltham MA and Joseph J. Rotman University of Illinois at Urbana–Champaign

Published and distributed by The Mathematical Association of America

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Committee on Books Frank Farris, Chair MAA Textbooks Editorial Board Zaven A. Karian, Editor Matthias Beck Richard E. Bedient Thomas A. Garrity Charles R. Hampton John Lorch Susan F. Pustejovsky Elsa J. Schaefer Stanley E. Seltzer Kay B. Somers

MAA TEXTBOOKS Bridge to Abstract , Ralph W. Oberste-Vorth, Aristides Mouzakitis, and Bonita A. Lawrence Deconstructed:A Second Coursein First-Year Calculus, Zbigniew H. Nitecki : A Guided Tour, David R. Mazur Combinatorics: A Problem Oriented Approach, Daniel A. Marcus Complex Numbers and , Liang-shin Hahn A Course in Mathematical Modeling, Douglas Mooney and Randall Swift Cryptological Mathematics, Robert Edward Lewand and its Applications, John Oprea Elementary Cryptanalysis, Abraham Sinkov Elementary Mathematical Models, Dan Kalman An Episodic : Mathematical Culture Through Problem Solving, Steven G. Krantz Essentials of Mathematics, Margie Hale Theory and its Classical Problems, Charles Hadlock Fourier Series, Rajendra Bhatia and Strategy, Philip D. Straffin Geometry Revisited, H. S. M. Coxeter and S. L. Greitzer : A Problem Oriented Approach, Daniel Marcus Knot Theory, Charles Livingston Learning Modern Algebra: From Early Attempts to Prove Fermat’s Last Theorem, Al Cuoco and and Joseph J. Rotman Lie Groups: A Problem-Oriented Introduction via Groups, Harriet Pollatsek Mathematical Connections: A Companion for Teachers and Others, Al Cuoco Mathematical Interest Theory, Second Edition, Leslie Jane Federer Vaaler and James W. Daniel

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Mathematical Modeling in the Environment, Charles Hadlock Mathematics for BusinessDecisionsPart 1: Probability and Simulation (electronic text- book), Richard B. Thompson and Christopher G. Lamoureux Mathematics for Business DecisionsPart 2: Calculus and Optimization (electronic text- book), Richard B. Thompson and Christopher G. Lamoureux Mathematics for Secondary School Teachers, Elizabeth G. Bremigan, Ralph J. Bremi- gan, and John D. Lorch The Mathematics of Choice, Ivan Niven The Mathematics of Games and Gambling, Edward Packel Math Through the Ages, William Berlinghoff and Fernando Gouvea Noncommutative Rings, I. N. Herstein Non-, H. S. M. Coxeter Through Inquiry, David C. Marshall, Edward Odell, and Michael Star- bird A Primer of Real Functions, Ralph P. Boas A Radical Approach to Lebesgue’s Theory of Integration, David M. Bressoud A Radical Approach to , 2nd edition, David M. Bressoud Real Infinite Series, Daniel D. Bonar and Michael Khoury, Jr. Now!, Robert Messer and Philip Straffin Understanding our Quantitative World, Janet Andersen and Todd Swanson

MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-301-206-9789

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vii

Per Micky: Tutto quello che faccio, lo faccio per te.

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Contents

Preface xiii SomeFeaturesofThisBook ...... xiv ANotetoStudents ...... xv ANotetoInstructors ...... xv

Notation xvii

1 Early Number Theory 1 1.1 AncientMathematics ...... 1 1.2 ...... 7 GeometryandPythagoreanTriples ...... 8 TheMethodofDiophantus ...... 11 Fermat’sLastTheorem ...... 14 Connections:CongruentNumbers...... 16 1.3 ...... 20 GreekNumberTheory...... 21 DivisionandRemainders ...... 22 LinearCombinationsandEuclid’sLemma ...... 24 EuclideanAlgorithm...... 30 1.4 NineFundamentalProperties ...... 36 1.5 Connections...... 41 ...... 41 Integration ...... 42

2 Induction 45 2.1 InductionandApplications ...... 45 UniqueFactorization...... 53 StrongInduction...... 57 DifferentialEquations ...... 60 2.2 BinomialTheorem ...... 63 Combinatorics ...... 69 2.3 Connections...... 73 AnApproachtoInduction ...... 73 FibonacciSequence ...... 75

3 Renaissance 81 3.1 ClassicalFormulas ...... 82 3.2 ComplexNumbers ...... 91 ix

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x Contents

AlgebraicOperations ...... 92 AbsoluteValueandDirection ...... 99 TheGeometryBehindMultiplication ...... 101 3.3 RootsandPowers ...... 106 3.4 Connections:DesigningGoodProblems ...... 116 Norms ...... 116 PippinsandCheese ...... 118 Gaussian : Pythagorean Triples Revisited ...... 119 EisensteinTriplesandDiophantus...... 122 NiceBoxes...... 123 NiceFunctionsforCalculusProblems ...... 124 LatticePointTriangles...... 126

4 Modular 131 4.1 Congruence ...... 131 4.2 PublicKeyCodes ...... 149 4.3 CommutativeRings...... 154 UnitsandFields ...... 160 SubringsandSubfields...... 166 4.4 Connections:JuliusandGregory ...... 169 4.5 Connections:PatternsinDecimalExpansions ...... 177 RealNumbers ...... 177 DecimalExpansionsofRationals ...... 179 PeriodsandBlocks...... 182

5 Abstract Algebra 191 5.1 DomainsandFractionFields ...... 192 5.2 ...... 196 PolynomialFunctions ...... 204 5.3 ...... 206 ExtensionsofHomomorphisms ...... 213 ,Image,andIdeals ...... 216 5.4 Connections:BooleanThings...... 221 Inclusion-Exclusion ...... 227

6 Arithmetic of Polynomials 233 6.1 Parallels to Z ...... 233 Divisibility...... 233 Roots...... 239 GreatestCommonDivisors ...... 243 UniqueFactorization...... 248 PrincipalIdealDomains ...... 255 6.2 Irreducibility ...... 259 RootsofUnity ...... 264 6.3 Connections:LagrangeInterpolation ...... 270

7 Quotients, Fields, and Classical Problems 277 7.1 QuotientRings ...... 277 7.2 FieldTheory...... 287 Characteristics ...... 287 ExtensionFields ...... 289

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Contents xi

AlgebraicExtensions ...... 293 SplittingFields...... 300 ClassificationofFiniteFields ...... 305 7.3 Connections:Ruler–CompassConstructions ...... 308 Constructing Regular n-gons ...... 320 Gauss’sconstructionofthe17-gon ...... 322

8 Cyclotomic Integers 329 8.1 ArithmeticinGaussianandEisensteinIntegers ...... 330 EuclideanDomains ...... 333 8.2 PrimesUpstairsandPrimesDownstairs ...... 337 LawsofDecomposition ...... 339 8.3 Fermat’sLastTheoremforExponent3 ...... 349 Preliminaries...... 350 TheFirstCase ...... 351 Gauss’sProofoftheSecondCase ...... 354 8.4 ApproachestotheGeneralCase ...... 359 Cyclotomicintegers ...... 360 Kummer,IdealNumbers,andDedekind...... 365 8.5 Connections:CountingSumsofSquares ...... 371 AProofofFermat’sTheoremonDivisors ...... 373

9 Epilog 379 9.1 AbelandGalois...... 379 9.2 SolvabilitybyRadicals ...... 381 9.3 ...... 384 9.4 Groups...... 389 9.5 WilesandFermat’sLastTheorem ...... 396 EllipticIntegralsandEllipticFunctions ...... 397 CongruentNumbersRevisited...... 400 EllipticCurves...... 404

A Appendices 409 A.1 Functions ...... 409 A.2 EquivalenceRelations...... 420 A.3 VectorSpaces ...... 424 BasesandDimension ...... 427 LinearTransformations ...... 435 A.4 Inequalities ...... 441 A.5 GeneralizedAssociativity...... 442 A.6 ACyclotomicIntegerCalculator ...... 444 EisensteinIntegers ...... 445 SymmetricPolynomials ...... 446 AlgebrawithPeriods...... 446

References 449

Index 451

About the Authors 459

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Preface

This book is designed for college students who want to teach mathematics in high school, but it can serve as a text for standard abstract algebra courses as well. First courses in abstract algebra usually number theory, groups, and commutative rings. We have found that the first encounter with groups is not only inadequate for future teachers of high school mathematics, it is also unsatisfying for other mathematics students. Hence, we focus here on number theory, polynomials, and commutative rings. We introduce groups in our last chapter, for the earlier discussion of commutative rings allows us to explain how groups are used to prove Abel’s Theorem: there is no generalization of the quadratic, cubic, and quartic formulas giving the roots of the general quintic . A modest proposal: undergraduate abstract algebra should be a sequence of two courses, with number theory and commutative rings in the first course, and groups and (with scalars in arbitrary fields) in the second. We invoke an historically accurate organizing principle: Fermat’s Last The- orem (in Victoriantimes, the title of this book wouldhave been Learning Mod- ern Algebra by Studying Early Attempts, Especially Those in the Nineteenth Century, that Tried to Prove Fermat’s Last Theorem Using Elementary Meth- ods). To be sure, another important problem at that time that contributed to modern algebra was the search for formulas giving the roots of polynomials. This search is intertwined with the algebra involved in Fermat’s Last Theo- rem, and we do treat this part of algebra as well. The difference between our approach and the standard approach is one of emphasis: the natural direction for us is towards , whereas the usual direction is to- wards . Four thousand years ago, the quadratic formula and the Pythagorean The- orem were seen to be very useful. To teach them to new generations, it was best to avoid roots (which, at the time, were complicated to compute), and so problems were designed to have solutions. This led to Pythag- orean triples: positive integers a;b;c satisfying a2 b2 c2. Two thousand years ago, all such triples were found and, when studyingC D them in the seven- teenth century, Fermat wondered whether there are positive integer solutions to an bn cn for n>2. He claimed in a famous marginal note that there are noC solutions,D but only his proof of the case n 4 is known. This problem, called Fermat’s Last Theorem, intrigued many ofD the finest mathematicians, but it long resisted all attempts to solve it. Finally, using sophisticated tech- niques of developed at the end of the twentieth century, proved Fermat’s Last Theorem in 1995.

xiii

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xiv Preface

Before its solution, Fermat’s Last Theorem was a challenge to mathemati- cians (as climbing Mount Everest was a challenge to mountaineers). There are no dramatic applicationsof the result, but it is yet another triumph of human in- tellect.What is trueis that,over thecourseof 350 years, much of contemporary mathematics was invented and developed in trying to deal with it. The num- ber theory recorded in Euclid was shown to have similarities with the behavior of polynomials, and generalizations of prime numbers and unique factoriza- tion owe their initial study to attempts at proving Fermat’s Last Theorem. But these topics are also intimately related to what is actually taught in high school. Thus, abstract algebra is not merely beautiful and interesting, but it is also a valuable, perhaps essential, topic for understanding high school mathematics.

Some Features of This Book We include sections in every chapter, called Connections, in which we explic- itly show how the material up to that point can help the reader understand and implement the mathematics that high school teachers use in their profession. This may include the many ways that results in abstract algebra connect with core high school ideas, such as solving or factoring. But it may also include mathematics for teachers themselves, that may or may not end up “on the blackboard;” things like the use of abstract algebra to make up good prob- lems, to understand the foundations of topics in the curriculum, and to place the topics in the larger landscape of mathematics as a scientific discipline. Many students studying abstract algebra have problems understanding proofs; even though they can follow each step of a proof, they wonder how anyone could have discovered its argument in the first place. To address such problems, we have tried to strike a balance between giving a logical develop- ment of results (so the reader can see how everything fits together in a coherent package) and discussing the messier kinds of thinking that lead to discovery and proofs. A nice aspect of this sort of presentation is that readers participate in doing mathematics as they learn it. One way we implement this balance is our use of several design features, such as the Connections sections described above. Here are some others. Sidenotes provide advice, comments, and pointers to other parts of the text  related to the topic at hand. What could be more fitting for a book related to Fermat’s Last Theorem than to have large margins? Interspersed in the text are boxed “callouts,” such as How to Think About  It, which suggest how ideas in the text may have been conceived in the first place, how we view the ideas, and what we guess underlies the formal exposition. Some other callouts are: Historical Note, which provides some historical background. It often helps to understand mathematical ideas if they are placed in historical con- text; besides, it’s interesting. The biographies are based on those in the MacTutor History of Mathematics Archive of the School of Mathemat- ics and , University of St. Andrews, Scotland. It can be found on the internet: its URL is www-history.mcs.st-andrews.ac.uk Etymology, which traces out the origin of some mathematical terms. We believe that knowing the etymology of terms often helps to understand the ideas they name.

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Preface xv

Etymology. The word mathematics comes from classical Greek; it means “knowledge,” “something learned.” But in ancient Rome through the thirteenth century, it meant “astronomy” and “astrology.” From the Middle Ages, it acquired its present meaning. The word arithmetic comes from the Greek word meaning “the art of counting.” The word geometry, in classical Greek, meant “science of measuring;” it arose from an earlier term meaning “land survey.”

It is a pleasure to acknowledge those who have contributed valuable com- ments, suggestions, ideas, and help. We thank Don Albers, Carol Baxter, Bruce Berndt, Peter Braunfeld, Keith Conrad, Victoria Corkery, Don DeLand, Ben Conrad’s website Fischer, Andrew Granville, Heini Halberstam, Zaven Karian, Tsit-Yuen Lam, www.math.uconn.edu/ Paul Monsky, Beverly Ruedi, Glenn Stevens, and Stephen Ullom. ˜kconrad/blurbs/ is full of beautiful ideas.

A Note to Students

The heart of a mathematics course lies in its problems. We have tried to or- chestrate them to help you build a solid understanding of the mathematics in the sections. Everything afterward will make much more sense if you work through as many exercises as you can, especially those that appear difficult. Quite often, you will learn something valuable from an exercise even if you don’t solve it completely. For example, a problem you can’t solve may show that you haven’t fully understood an idea you thought you knew; or it may force you to discover a fact that needs to be established to finish the solution. There are two special kinds of exercises. Those labeled Preview may seem to have littleto do withthesection at hand;  they are designed to foreshadow upcoming topics, often with numerical ex- periments. Those labeled Take it Further develop interesting ideas that are connected  to the main themes of the text, but are somewhat off the beaten path. They are not essential for understanding what comes later in the text. An exercise marked with an asterisk, such as 1.8*, means that it is either used in some proof or it is referred to elsewhere in the text. For ease of finding such exercises, all references to them have the form “Exercise 1.8 on page 6” givingboth its number and the number of the page on which it occurs.

A Note to Instructors

We recommend giving reading assignments to preview upcoming material. This contributesto balancing experience and formality as described above, and it saves time. Many important pages can be read and understood by students, and they shouldbe discussed inclass onlyif studentsask questions about them. It is possible to use thisbook as a text for a three hour one-semester course, but we strongly recommend that it be taught four hours per week.

—Al Cuoco and Joe Rotman

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Notation

.a; b; c/ 4 triangle with sides of lengths a;b;c ABC 4 triangle with vertices A; B; C N 21 natural numbers Z 21 integers a b 21 a is a divisor of b j gcd.a; b/ 24 greatest common divisor x 29 greatest integer in x b c Q 36 rational numbers R 36 real numbers 46 implies ) lcm.a; b/ 55 least common multiple n r 63 binomial coefficient .z/ 92 real part of z <  .z/ 92 imaginary part of complex number z = C 92 complex numbers !PQ 93 arrow from P to Q z 96 conjugate of z z 99 modulus of z j j arg.z/ 100 argument of z ez 108 complex exponential .n/ 111 Euler -function N.z/ 116 of z ZŒi 119 Gaussian integers ZŒ!/ 120 Eisenstein integers a b mod m 132 a is congruent to b modulo m Á m1 mi mr 147 expression with mi deleted   Œa 154 congruence class of integer a

Zcm 154 integers mod m ZŒ 157 cyclotomic integers RS 157 of functions R S ! C.X/ 157 ring of continuous functions X R ! xvii

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xviii Notation

Fun.R/ 157 ring of functions R R ! F4 165 field with 4 elements 2X 167 Boolean ring of subsets of X j.m/ 172 calendar month function Frac.D/ 194 fraction field of domain D a=b 195 element of Frac.D/ deg.f / 198 degree of polynomial f RŒŒx 198 all over R RŒx 198 all polynomials over R x 200 in RŒx f .x/ 202 derivative of f .x/ RŒx 0 2 f # 204 associated polynomial function of f Poly.R/ 204 all polynomials functions over R k.x/ 205 field of rational functions over k

Fq 205 finite field with exactly q elements

RŒx1;:::;xn 205 polynomials in several variables over R

D.x1;:::;xn/ 206 rational functions in several variables over domain D R S 207 rings R and S are isomorphic Š ker ' 217 kernel of ' im ' 217 image of homomorphism '

.b1;:::;bn/ 218 generated by b1;:::;bn .a/ 218 principal ideal generated by a .0/ 219 zero ideal 0 Df g IJ 220 product of ideals I and J I J 220 sum of ideals I and J C R S 221 direct product of rings R and S  a b 223 in Boolean ring _ A 227 number of elements in finite set A j j PID 255 principal ideal domain UFD 258 unique factorization domain

ˆd .x/ 265 cyclotomic polynomial a I 278 coset of element a mod ideal I C a b mod I 279 congruent mod ideal I Á R=I 280 R mod I X 293 subfield generated by subset X ŒK k 291 degree of field K=k ˝ W ˛ k.z1 ;:::;zn/ 294 extension field adjoining z1;:::;zn to k irr.z; k/ 296 minimal polynomial of z over k PQ 310 with endpoints P; Q PQ 310 length of segment PQ

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Notation xix

L.P; Q/ 309 line determined by points P; Q C.P; Q/ 309 circle with center P , radius PQ @ 333 size function on Euclidean domain  348  1 ! D  350 valuation r.n/ 371 number of non-associate z ZŒi of norm n 2 Q1 372 first quadrant .s/ 374 Riemann zeta function .n/ 375 a multiplicative function on ZŒi Gal.f / 386 Galois of polynomial f Gal.E=k/ 387 of field extension E=k

Sn 389 on n letters G=N 392 quotient group a A 409 a is an element of set A 2 1X 411 identity function on set X f a b 411 f .a/ b W 7! D U V 410 U is a subset of set V  U ¨ V 410 U is a proper subset of V ¿ 410 empty set g f 414 composite f followed by g ı Œa 421 of element a Span X 427 subspace spanned by subset X h i dim.V / 433 dimension of V

V  437 dual space of vector space V

A> 438 transpose of matrix A

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References

[1] Apostol, Tom M., Introduction to , Under- graduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. [2] Artin, E., Galois Theory, Edwards Brothers, Ann Arbor, 1948. [3] Baker, A., Transcendental Number Theory, 2d ed., Cambridge Univer- sity Press, Cambridge, 1990. [4] Barbeau, E. J., Polynomials, Springer-Verlag, New York, 1989. [5] Borevich, Z. I., and Shafarevich, I. R., Number Theory, Academic Press, New York, 1966. [6] Cajori, F., A History of Mathematical Notation, Open Court, 1928; Dover reprint, Mineola NY, 1993. [7] Cuoco, A., Mathematical Connections, Classroom Resource Materials, MAA, Washington, 2005. [8] Dirichlet, P. G. L., Lectures on Number Theory, Supplements by R. Dedekind. Translated from the 1863 German original and with an intro- duction by John Stillwell. History of Mathematics, 16. American Math- ematical Society, Providence; London Mathematical Society, London, 1999. [9] Du Val, P., Elliptic Functions and Elliptic Curves, London Mathemati- cal Society Lecture Note Series 9, Cambridge University Press, London, 1973. [10] EDC, The CME Project, Pearson, Boston, 2013 [11] Edwards, H. M., Fermat’s Last Theorem. A Genetic Introduction to Number Theory, Graduate Texts in Mathematics vol. 50, Springer- Verlag, New York, 1977. [12] Euler, L., Elements of Algebra, Translated from the German by John Hewlett. Reprint of the 1840 edition. With an introduction by C. Trues- dell, Springer-Verlag, New York, 1984. [13] Flannery, S., In Code. A Mathematical Journey, Reprint of the 2000 original, Workman Publishing, New York, 2001. [14] Gauss, C. F., Disquisitiones Arithmeticae, Arthur A. Clarke (trans), Yale University Press, New Haven, 1966. [15] Hadlock, C. R., Field Theory and Its Classical Problems, Carus Mathe- matical Monographs 19, Mathematical Association of America, Wash- ington, 1978. 449

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450 References

[16] Heath, T. L., The Thirteen Books of Euclid’s Elements, Cambridge Uni- versity Press, 1926; Dover reprint, Mineola NY, 1956. [17] Ireland, K. and Rosen, M., A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, 1982. [18] Koblitz, N., A Course in Number Theory and , Springer- Verlag, New York, 1987. [19] ——, Introductionto EllipticCurves and Modular Forms, Springer Ver- lag, New York, 1993. [20] Loomis, E. S., The Pythagorean Proposition, Edwards Brothers., Ann Arbor, 1940. [21] Montgomery, S., and Ralston, E. W., Selected Papers in Algebra, Ray- mond W. Brink Selected Mathematical Papers, Vol. 3, Mathematical As- sociation of America, Washington, 1977. [22] Needham, T., Visual , Clarendon Press, Oxford, 1997. [23] Ribenboim, P., Thirteen Lectures on Fermat’s Last Theorem, Springer– Verlag, New York, 1979. [24] Rosen, K. H., Elementary Number Theory and Its Applications, 4th ed., Addison-Wesley, Reading, MA, 2000. [25] Rotman, J. J., Advanced Modern Algebra, 2d ed., Graduate Studies in Mathematics vol. 114, American Mathematical Society, Providence, 2010. [26] ——, A First Course in Abstract Algebra, 3d ed., Prentice Hall, Upper Saddle River NJ, 2006. [27] ——, Galois Theory, 2d ed., Springer-Verlag, New York, 1998. [28] ——, Journey into Mathematics, Prentice Hall, Upper Saddle River NJ, 1998; Dover reprint, Mineola NY, 2007. [29] Samuel, P., Algebraic Theory of Numbers, Houghton-Mifflin, Boston, 1992. [30] Siegel, C. L., Topics in Complex Function Theory Vol. I, Elliptic Func- tions and Uniformization Theory, Wiley, New York, 1969. [31] Silverman, J. H., and Tate, J., Rational Points on Elliptic Curves, Springer–Verlag, New York, 1992. [32] Stillwell, J., Mathematics and Its History, 3d ed., Springer, New York, 2010. [33] Tignol, J.-P., Galois’ Theory of Equations, World Scientific Publishing, Singapore, 2001. [34] van der Waerden, B. L., Science Awakening, John Wiley, New York, 1963. [35] ——, Geometry and Algebra in Ancient Civilizations, Springer–Verlag, New York, 1983. [36] Washington, L.C., Introduction to Cyclotomic Fields, Springer, New York, 1982. [37] Weil, A., Number Theory: An Approach Through History. From Ham- murapi to Legendre, Birkh¨auser, Boston, 1984. [38] Weyl, H., Algebraic Theory of Numbers, 6th printing,Princeton Univer- sity Press, Princeton, 1971; Princeton Landmarks in Mathematics and Physics Series, 1998. [39] Wilf, H., Generatingfunctionology, Academic Press, New York, 1994.

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Index

Abel, Niels H., 379, 391, 394, 400 Braunfeld, Peter, 53 , 390 Theorem, 399 calendar adjoining to field, 294 formula Adleman, Leonard M., 150 Conway, John H., 175 al-Khwarizmi, 240 Gregorian, 173 algebraic Julian, 169 element, 293 Mayan, 144 extension field, 293 cancellation law, 192 algebraic , 254 Cardano, Girolamo, 81 algebraically closed, 301 cartesian product, 410 anagram, 70 CAS = System, 142 antanairesis, 31 casting out 9s, 137 , 321 castle problem, 90 Aristarchus, 3 Casus Irreducibilis, 383 Arithmetic–Geometric Mean Inequal- century year, 170 ity, 6 characteristic function, 209 Artin, Emil, 386 characteristic of field, 288 associate, 234 Ch’in Chiu-shao = Qin Jiushao, 90, associated polynomial function, 204 146 associativity, 37 Chinese Remainder Theorem, 142 generalized, 443 –tangent construction, 405 atom, 227 closed under operation, 155 , 386 coconuts, 148 code (see RSA code), 150 b-adic digits, 139 coefficients, 197 Babylonian method, 5 commensurable, 28 base b, 139 common divisor, 24 basis in Z, 24 standard, 430 polynomials, 243 vector space, 430 common multiple Bernoulli, Jacob, 398 in Z, 55 Bernoulli, James, 398 polynomials, 253 Bhaskara I, 149 common year, 169 bijective, bijection, 416 commutative, 37 binary operation, 155 , 156 Binomial Theorem formal power series, 200 in Z, 66 in commutative ring, 160 one variable, 200 Boolean ring, 167, 223 several variables, 205 451

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452 Index

compass, 309 degree, 198 complement, 168 extension field, 291 completing the square, 2, 23 derivative, 202 complex conjugate, 96 Descartes, Ren´e, 82 complex exponential, 108 diagonal, 420 complex number, 92 dimension, 433 absolute value, 99 Diophantine , 404 argument, 100 Diophantus, 5, 8, 12, 27, 405 conjugate, 96 direct product, 221 exponential form, 109 of vector spaces, 440 imaginary part, 92 Dirichlet, J. P. G. Lejeune, 153, 358, modulus, 99 360 norm, 116 discriminant, 91 polar form, 101 disjoint, 422 real part, 92 distance preserving, 385 composite distributive, 37 functions, 414 divides, 21 number, 22 Algorithm congruence class, 154, 421 integers, 23 congruent mod I , 278 polynomials, 237 congruent mod m, 132, 270 divisor, 21 congruent numbers, 400 commutative ring, 233 constant function, 411 proper, 257 constant polynomial, 201 divisors, 368 constant term, 201 domain, 192 constructible, 311 Euclidean domain, 333 number, 312 of function, 410 point, 311 doomsday, 174 subfield, 313 double formula, 107 contrapositive, 26 dual basis, 438 convex, 48 dual space, 437 Conway, John H., 175 coordinate list, 431 Eisenstein integers, 120 coset Eisenstein Criterion, 267 ideal, 279 Eisenstein triple, 121 subgroup, 392 Eisenstein, F. Gotthold M., 120, 267 Cramer’s Rule, 164 Elements, 2, 20 Crelle, August L., 380 elliptic curve, 404 cross multiplication, 193 elliptic function, 400 cubic formula, 84 elliptic integral, 398 cubic polynomial, 382 empty set, 410 cyclotomic integers, 157 equality of functions, 412 cyclotomic polynomial, 265 equality of sets, 410 equivalence class, 421 day, 169 equivalence relation, 420 de Moivre’s Theorem, 107 Erlanger Programm, 390 exponential form, 109 etymology de Moivre, Abraham, 107 abelian, 391 De Morgan, Augustus, 169 algebra, 240 decimal expansion, 177 algorithm, 240 Dedekind, Richard, 218, 366, 369 arithmetic, xv

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Index 453

binomial, 63 degree, 291 calculus, 197 finite, 291 calendar, 173 pure, 382 casting out 9s, 137 radical, 383 coefficients, 198 corollary, 11 Factor Theorem, 240 cubic, 202 factorial, 51 degree Fagnano, Giulio, 399 field extension, 291 Fermat prime, 325 polynomial, 198 Fermat’s n 4 Theorem, 15 geometry, xv Fermat’s LastD Theorem, 14 , 76 Fermat’s Little Theorem, 136 homomorphism, 207 Fermat’s Theorem on Divisors, 372 hypotenuse, 4 Fermat’s Two-Square Theorem, 342 ideal, 218, 366 Fermat, Pierre de, 14 , 207 Ferrari, Ludovico, 87 lemma, 11 Fibonacci sequence, 75 linear, 202 Fibonacci, Leonardo, 19, 75 mathematics, xv field, 163 modulo, 132 algebraically closed, 301 monomial, 63 extension, 291 power, 51 finite, 305 proof, 11 prime, 293 proposition, 10 rational functions, 205 quadratic, 202 splitting, 301 radical, 240 ring, 157 finite extension field, 291 root, 239 finite-dimensional, 428 scalar, 425 first day of month, 173 September, 171 first form of induction, 57 theorem, 11 First Isomorphism Theorem vector, 424 commutative rings, 282 Euclid, 2, 20, 23 groups, 392 Elements, 20 first quadrant, 372 Euclid’s Lemma fixes, 386 integers, 25 formal power series, 197 PID, 256 fraction field, 194 polynomials, 248 Frey, Gerhard, 407 I function, 410 integers, 32 bijective, 416 polynomials, 249 constant, 411 Euclidean Algorithm II identity, 411 integers, 32 inclusion, 412 polynomials, 250 injective, 413 Eudoxus, 28, 308 restriction, 415 Euler -function, 111, 232 surjective, 413 Euler, Leonhard, 108, 111, 131, 326, functional, 437 330, 399 Fundamental Theorem evaluation homomorphism, 215 Algebra, 105 extension field, 291 Arithmetic, 54 algebraic, 293 Galois Theory, 393

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454 Index

Galois field, 306 homomorphism, 217 Galois group, 386 inclusion, 412 Galois, Evariste,´ 305, 306, 380, 394 Inclusion–Exclusion, 230 Gauss’s Lemma, 260 independent list Gauss, Carl F., 50, 258, 268, 321, 380, longest, 434 384, 400 indeterminate, 200 Gauss-Wantzel theorem, 325 indirect proof, 27 Gaussian integers, 7, 119, 157 induction gcd, 24, 243, 255 base step, 47 Generalized Associativity, 443 first form, 47 Germain, Marie-Sophie, 358 inductive hypothesis, 47 golden ratio, 76 inductive step, 47, 57 googol, 140 strong, 57 Granville, Andrew J., 406 inductive reasoning, 45 graph of function, 411 inequalities, 441 greatest common divisor,24, 243, 255 inert, 342 greatest integer function, 29 infinite descent, 12 Gregorian calendar, 170 infinite-dimensional, 428 group, 389 injective, injection, 413 abelian, 390 integers mod m, 154 quotient, 392 integers Z, 21 solvable, 394 rational, 339 symmetric, 389 integers, cyclotomic, 359 integration, 42 Heath, Thomas L., 2, 20 invariance of dimension, 433 Helikon, 308 Heron triangle, 14 additive, 37 Hilbert, David, 157, 369 group, 389 hockey stick, 68 multiplicative, 37, 156 H¨older, Otto L., 383 inverse function, 416 Holmes, Sherlock, 176 irr.z; k/, 296 homomorphism, 207 irreducible in commutative ring, 234 group, 391 isometry, 385 Hume, James, 82 isomorphism, 207 Hungerb¨uhler, Norbert, 320 group, 391 hyperbolic cosine, 6 Jacobi, Carl G. J., 360, 380, 400 ideal, 218 Joachimsthal, Ferdinand, 359 generated by X, 168 Julian calendar, 169 maximal, 286 prime, 286 k-linear combination, 427 principal, 218 Kaplansky, Irving, 236 product, 220 kernel, 217 proper, 218 group, 392 sum, 220 Klein, Felix C., 390 identity element Kolyvagin, Victor, 407 commutative ring, 156 Kronecker, Leopold, 28, 300, 359 group, 389 Kummer, Eduard, 218, 358, 359 identity function, 411 image Lagrange Interpolation, 272 function, 410 Lagrange, Joseph-Louis, 272

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Index 455

Lam´e, Gabriel, 32, 358 natural , 282 point, 13, 35 natural numbers N, 21 Law of Decomposition n choose r, 63 Eisenstein integes, 346 negative, 37, 156 Gaussian integers, 343 Newton, Isaac, 197, 405 Law of Substitution, 37, 155, 412 Nine Chapters on theMathematical Art, Laws of Exponents, 52 3, 6 leading coefficient, 198 Noether, Emmy, 369 leap year, 169 norm, 116 least common multiple , 392 in Z, 55 polynomials, 253 one-one correspondence Least Integer , 21 see bijective, 416 Legendre, Adrien-Marie, 358 one-to-one Leibniz, Gottfried W., 68, 397 see injective, 413 Levi ben Gershon, 52 onto (function) lies above, 340 see surjective, 413 Lindemann, C. L. Ferdinand von, 293 order of power series, 203 linear combination, 218 order of unit, 165 in Z, 24 ordered pair, 410 vector space, 427 Oresme, Nicole, 45 linear transformation, 435 origin, 310 linearly dependent, 429 Oughtred, William, 82 linearly independent, 429, 430 pairwise disjoint, 422 Liouville, Joseph, 381, 397 law, 93, 425 list, 427 , 131 longest independent list, 434 partition, 422 lowest terms Pascal’s triangle, 63 in Q, 26 Pascal, Blaise, 52, 64 rational functions, 248 period, 179 , 385 Mascheroni, Lorenzo, 320 PID, 255 matrix associated to linear transforma- Pigeonhole Principle, 419 tion, 437 Plato, 308 Maurolico, Francesco, 52 Plimpton 322, 4 maximal ideal, 286 Pogrebishte, 176 Mayan calendar, 144 pointwise addition, 157 Mazur, Barry, 407 pointwise multiplication, 157 minimal polynomial, 296 polar form, 101 M¨obius, August F., 263 polynomial modulus, 132 n variables, 205 Mohr, Georg, 320 monic, 201 monic polynomial, 201 one variable, 197 Moore, Eliakim H., 306 reduced, 83 Motzkin, Theodore S., 335 splits, 301 multiple, 21 zero, 198 multiplicative function, 378 polynomial function, 204 strongly multiplicative function, polynomial ring 375 one variable, 200 multiplicity, 254 several variables, 205

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456 Index

power series, 197 rational powers, 51 block, 179 commutative ring, 160 function, 205 predecessor, 57 integer, 339 prime, 22 line, 13 inert, 342 period, 179 ramifies, 342 point, 10 rational, 339 prime, 339 splits, 340 repeats, 179 prime factorization, 54, 252 terminates, 179 prime field, 293 Rational Root Theorem, 260 prime ideal, 286 Recorde, Robert, 82 primes recurrence, 74 irregular, 368 reduced polynomial, 83 regular, 368 reduction mod p, 260 primitive element, 325 reflexive relation, 420 primitive Pythagorean triple, 27 relation, 420 primitive root of unity, 111, 264 relatively prime principal ideal, 218 integers, 26 principal ideal domain, 255 polynomials, 248 private key, 150 remainder, 23 , 20 Remainder Theorem, 240 product expansion, 375 repeated roots, 263 proof resolvent cubic, 87, 384 by contradiction, 27 restriction, 415 indirect, 27 Ribet, Kenneth A., 407 proper Riemann zeta function, 374 ideal, 218 Riemann, G. F. Bernhard, 369 subset, 410 ring, commutative, 156 subspace, 426 Rivest, Ronald L., 150 public key, 150 root pure extension, 382 multiplicity, 254 , 2 polynomial, 81, 239 Pythagorean root of unity, 111 point, 10 commutative ring, 165 Theorem, 2 cube root, 82 converse, 7 primitive, 111, 264 triple, 4 RSA code, 150 primitive, 27 Ruffini, Paolo, 394 similar, 9 ruler, 309

Qin Jiushao, 90, 146 scalar, 424 quadratic formula, 2 multiplication, 424 quartic formula, 87 Sch¨onemann, Theodor, 267 quartic polynomial, 383 Scherk, Heinrich F., 359 quotient, 23, 38 Scipione, 81 quotient group, 392 Serre, Jean-Pierre, 407 quotient ring, 281 Shamir, Adi, 150 Shimura, Goro, 397, 407 radical extension, 383 similar, 9 Rahn, Johann H., 82 Singer, Richard, 267

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Index 457

single-valued, 412 Uncle Charles, 174 size function, 333 unique, 23 smallest subspace, 427 unique factorization solution of linear system, 426 in Z, 54 solution space, 426 polynomials, 252 solvable by radicals, 383 unique factorization domain, 258 solvable group, 394 unit, 162 spans, 427 unit n-gon, 110 splits unit circle, 8 polynomial, 301 unit vector, 99 prime, 340 splitting field, 301 valuation, 350 squarefree integer, 34 value of function, 410 standard basis, 430 van der Waerden, Bartel L., 21, 28 straightedge (see ruler), 309 Vandermonde’s Identity, 73 strong induction, 57 Vandermonde, Alexandre, 73 subfield, 167 vector space, 424 generated by X, 293 vectors, 424 subgroup, 390 Venn diagram, 222 normal, 392 Venn, John, 222 subring, 166 Vi`ete, Francois, 82, 88 generated by X , 168 Wantzel, Pierre L., 318 subset, 410 Weierstrass, Karl, 404 subspace, 426 Weil, Andr´e, 407 proper, 426 well-defined, 412 smallest, 427 Well-Ordering Axiom, 21 spanned by X, 427 Wessel, Caspar, 92 , 38 Widman, Johannes, 82 surjective, 413 Wiles, Andrew J., 14, 396, 407 surjective, surjection, 413 Williams, Kenneth S., 335 symmetric difference, 166 Wilson’s Theorem, 263 symmetric group, 389, 418 Wilson, Jack C., 335 symmetric relation, 420 Wilson, John, 263 symmetry, 385 year, 169 tangent half-angle formula, 43 century year, 170 Taniyama, Yutaka, 397, 407 common, 169 target, 410 leap year, 169 Tartaglia = Nicolo Fontana, 81 transcendental element, 293 zero polynomial, 198 transitive relation, 420 zero ring, 159 transpose, 438 , 192 triangular number, 68 trichotomy, 441 trigonometric identities, 42 triple angle formula, 110 Trotsky, , 170 tzolkin calendar, 144

UFD, 258 Uncle Ben, 176

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About the Authors

Al Cuoco is DistinguishedScholar and Director of the Center for at Education Development Center. He is lead author for The CME Project, a four-year NSF-funded high school curriculum, published by Pear- son. He also co-directs Focus on Mathematics, a mathematics-science part- nership that has established a mathematical community of mathematicians, teachers, and mathematics educators. The partnership evolved from his25-year collaboration with Glenn Stevens (BU) on Boston University’s PROMYS for Teachers, a professional development program for teachers based on the Ross program (an immersion experience in mathematics). Al taught high school mathematics to a wide range of students in the Woburn, Massachusetts public schools from 1969 until 1993. A student of Ralph Greenberg, Cuoco holds a Ph.D. from Brandeis, with a thesis and research in Iwasawa theory. In addition to this book, MAA publishedhis MathematicalConnections: a Companion for Teachers and Others. But his favorite publication is a 1991 paper in the Amer- ican Mathematical Monthly, described by his wife as an attempt to explain a number system that no one understands with a picture that no one can see.

Joseph Rotman was born in Chicago on May 26, 1934. He studied at the University of Chicago, receiving the degrees BA, MA, and Ph.D. there in 1954, 1956, and 1959, respectively; his thesis director was Irving Kaplansky. Rotman has been on the faculty of the mathematics department of the Uni- versity of Illinois at Urbana-Champaign since 1959, with the following ranks: Research Associate 1959–1961; Assistant Professor 1961–1963;Associate Pro- fessor 1963–1968; Professor 1968–2004; Professor Emeritus 2004–present. He has held the following visiting appointments: Queen Mary College, Lon- don, England 1965, 1985; Aarhus University, Denmark, Summer 1970; He- brew University, Jerusalem, Israel 1970; University of Padua, Italy, 1972; Tech- nion, Israel Institute of Technology and Hebrew University, Jerusalem (Lady Davis Professor), 1977–78; Tel Aviv University, Israel, 1982; Bar Ilan Univer- sity, Israel, Summer 1982; Annual visiting lecture, South African Mathemati- cal Society, 1985; Oxford University, England, 1990. Professor Rotman was an editor of Proceedings of American Mathematical Society, 1970, 1971; managing editor, 1972, 1973. Aside from writing research articles, mostly in algebra, he has written the following textbooks: 1965, 1973, 1984, 1995; Homological Al- gebra 1970, 1979, 2009; 1988; Galois Theory 1990, 1998; Journey into Mathematics 1998, 2007; First Course in Abstract Algebra 1996, 2000, 2006; Advanced Modern Algebra 2002.

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i i AMS / MAA TEXTBOOKS

Learning Modern Algebra aligns with the CBMS Mathematical Education of Teachers II recommendations, in both content and practice. It emphasizes rings and fields over groups, and it makes explicit connections between the ideas of abstract algebra and the mathematics used by high school teachers. It provides opportunities for prospective and practicing teachers to experience mathematics for themselves, before the formalities are developed, and it is explicit about the mathematical habits of mind that lie beneath the definitions and theorems. This book is designed for prospective and practicing high school mathematics teachers, but it can serve as a text for standard abstract algebra courses as well. The presentation is organized historically: the Babylonians introduced Pythagorean triples to teach the ; these were classified by Diophantus, and eventually this led Fermat to conjecture his Last Theorem. The text shows how much of modern algebra arose in attempts to prove this; it also shows how other important themes in algebra arose from questions related to teaching. Indeed, modern algebra is a very useful tool for teachers, with deep connections to the actual content of high school mathematics, as well as to the mathematics teachers use in their profession that doesn’t neces- sarily “end up on the blackboard.” The focus is on number theory, polynomials, and commutative rings. Group theory is introduced near the end of the text to explain why generalizations of the quadratic formula do not exist for polynomials of high degree, allowing the reader to appreciate the more general work of Galois and Abel on roots of polynomials. Results and proofs are motivated with specific examples whenever possible, so that abstractions emerge from concrete experience. Applications range from the theory of repeating decimals to the use of imag- inary quadratic fields to construct problems with rational solutions. While such applications are integrated throughout, each chapter also contains a section giving explicit connections between the content of the chapter and high school teaching.