Learning Modern Algebra: From Early Attempts to Prove Fermat’s Last Theorem

front cover
......Page 1
copyright page
......Page 3
title page
......Page 4
Contents......Page 10
Preface......Page 14
Some Features of This Book......Page 15
A Note to Instructors......Page 16
Notation......Page 18
Ancient Mathematics......Page 22
Diophantus......Page 28
Geometry and Pythagorean Triples......Page 29
The Method of Diophantus......Page 32
Fermat's Last Theorem......Page 35
Connections: Congruent Numbers......Page 37
Euclid......Page 41
Greek Number Theory......Page 42
Division and Remainders......Page 43
Linear Combinations and Euclid's Lemma......Page 45
Euclidean Algorithm......Page 51
Nine Fundamental Properties......Page 57
Trigonometry......Page 62
Integration......Page 63
Induction and Applications......Page 66
Unique Factorization......Page 74
Strong Induction......Page 78
Differential Equations......Page 81
Binomial Theorem......Page 84
Combinatorics......Page 90
An Approach to Induction......Page 94
Fibonacci Sequence......Page 96
Renaissance......Page 102
Classical Formulas......Page 103
Complex Numbers......Page 112
Algebraic Operations......Page 114
Absolute Value and Direction......Page 120
The Geometry Behind Multiplication......Page 122
Roots and Powers......Page 127
Connections: Designing Good Problems......Page 137
Norms......Page 138
Pippins and Cheese......Page 139
Gaussian Integers: Pythagorean Triples Revisited......Page 140
Eisenstein Triples and Diophantus......Page 143
Nice Boxes......Page 144
Nice Functions for Calculus Problems......Page 145
Lattice Point Triangles......Page 147
Congruence......Page 152
Public Key Codes......Page 170
Commutative Rings......Page 175
Units and Fields......Page 181
Subrings and Subfields......Page 187
Connections: Julius and Gregory......Page 190
Real Numbers......Page 198
Decimal Expansions of Rationals......Page 200
Periods and Blocks......Page 203
Abstract Algebra......Page 212
Domains and Fraction Fields......Page 213
Polynomials......Page 217
Polynomial Functions......Page 225
Homomorphisms......Page 227
Extensions of Homomorphisms......Page 234
Kernel, Image, and Ideals......Page 237
Connections: Boolean Things......Page 242
Inclusion-Exclusion......Page 248
Divisibility......Page 254
Roots......Page 260
Greatest Common Divisors......Page 265
Unique Factorization......Page 269
Principal Ideal Domains......Page 276
Irreducibility......Page 280
Roots of Unity......Page 285
Connections: Lagrange Interpolation......Page 291
Quotient Rings......Page 298
Characteristics......Page 308
Extension Fields......Page 310
Algebraic Extensions......Page 314
Splitting Fields......Page 321
Classification of Finite Fields......Page 326
Connections: Ruler--Compass Constructions......Page 329
Constructing Regular n-gons......Page 342
Gauss's construction of the 17-gon......Page 344
Cyclotomic Integers......Page 350
Arithmetic in Gaussian and Eisenstein Integers......Page 351
Euclidean Domains......Page 355
Primes Upstairs and Primes Downstairs......Page 358
Laws of Decomposition......Page 360
Fermat's Last Theorem for Exponent 3......Page 370
Preliminaries......Page 371
The First Case......Page 372
Gauss's Proof of the Second Case......Page 375
Approaches to the General Case......Page 380
Cyclotomic integers......Page 381
Kummer, Ideal Numbers, and Dedekind......Page 386
Connections: Counting Sums of Squares......Page 392
A Proof of Fermat's Theorem on Divisors......Page 394
Abel and Galois......Page 400
Solvability by Radicals......Page 402
Symmetry......Page 405
Groups......Page 410
Wiles and Fermat's Last Theorem......Page 417
Elliptic Integrals and Elliptic Functions......Page 418
Congruent Numbers Revisited......Page 421
Elliptic Curves......Page 425
Functions......Page 430
Equivalence Relations......Page 441
Vector Spaces......Page 445
Bases and Dimension......Page 448
Linear Transformations......Page 456
Inequalities......Page 462
Generalized Associativity......Page 463
A Cyclotomic Integer Calculator......Page 465
Eisenstein Integers......Page 466
Algebra with Periods......Page 467
References......Page 470
Index......Page 472
About the Authors......Page 480