Introduction to Modern Algebra - Clark U
Introduction to Modern Algebra
David Joyce Clark University Version 1.2.7, 5 Dec 2017 1
1Copyright (C) 2008,2017.
ii
I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write this book. I'm sorry that he did not live to see it finished.
Arthur was born in 1954 in Taipei, Taiwan. He received his bachelors in mathematics in 1976 from Tunghai University and his PhD from Stony Brook in 1982. After a year at the Institute for Advanced Study at Princeton, he joined Clark University in 1983. He was promoted to Associate Professor six years later and promoted to full professor in 2008, the year he died. Besides mathematics, he had many other interests. Among other things, he was the general manager of the North America Elite Youth Orchestra which performed at Dallas, Beijing, and Taipei, and he was the deacon of the Chinese Gospel Church in Southborough, Massachusetts.
Contents
List of Figures
vi
List of Tables
viii
1 Introduction
1
1.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Structures in Modern Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Operations on sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.4 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.5 Other algebraic structures besides fields, rings, and groups . . . . . . . 10
1.3 Isomorphisms, homomorphisms, etc. . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.3 Monomorphisms and epimorphisms . . . . . . . . . . . . . . . . . . . . 13
1.3.4 Endomorphisms and automorphisms . . . . . . . . . . . . . . . . . . . 14
1.4 A little number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.1 Mathematical induction on the natural numbers N . . . . . . . . . . . 15
1.4.2 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.3 Prime numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.4 The Euclidean algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5 The fundamental theorem of arithmetic . . . . . . . . . . . . . . . . . . . . . . 22
1.6 Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.6.1 Division for polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.6.2 Roots of unity and cyclotomic polynomials . . . . . . . . . . . . . . . . 28
2 Fields
31
2.1 Introduction to fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1.1 Definition of fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1.2 Subtraction, division, multiples, and powers . . . . . . . . . . . . . . . 32
2.1.3 Properties that follow from the axioms . . . . . . . . . . . . . . . . . . 33
2.1.4 Subfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.1.5 Fields of rational functions . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.6 Vector spaces over arbitrary fields . . . . . . . . . . . . . . . . . . . . . 35
2.2 Cyclic rings and finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
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CONTENTS
2.2.1 The cyclic ring Zn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.2 The cyclic prime fields Zp . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2.3 Characteristics of fields, and prime fields . . . . . . . . . . . . . . . . . 41 2.3 Field Extensions, algebraic fields, the complex numbers . . . . . . . . . . . . . 41 2.3.1 Algebraic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3.2 The field of complex numbers C . . . . . . . . . . . . . . . . . . . . . . 43 2.3.3 General quadratic extensions . . . . . . . . . . . . . . . . . . . . . . . . 44 2.4 Real numbers and ordered fields . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.4.1 Ordered fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.4.2 Archimedean orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.4.3 Complete ordered fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.5 Skew fields (division rings) and the quaternions . . . . . . . . . . . . . . . . . 50 2.5.1 Skew fields (division rings) . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.5.2 The quaternions H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 Rings
55
3.1 Introduction to rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.1 Definition and properties of rings . . . . . . . . . . . . . . . . . . . . . 55
3.1.2 Products of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1.3 Integral domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1.4 The Gaussian integers, Z[i] . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1.5 Finite fields again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Factoring Zn by the Chinese remainder theorem . . . . . . . . . . . . . . . . . 60 3.2.1 The Chinese remainder theorem . . . . . . . . . . . . . . . . . . . . . . 60
3.2.2 Brahmagupta's solution . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2.3 Qin Jiushao's solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Boolean rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3.1 Introduction to Boolean rings . . . . . . . . . . . . . . . . . . . . . . . 64
3.3.2 Factoring Boolean rings . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3.3 A partial order on a Boolean ring . . . . . . . . . . . . . . . . . . . . . 66
3.4 The field of rational numbers, fields of fractions . . . . . . . . . . . . . . . . . 67
3.5 Categories and the category of rings . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5.1 The formal definition of categories . . . . . . . . . . . . . . . . . . . . . 70
3.5.2 The category R of rings . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5.3 Monomorphisms and epimorphisms in a category . . . . . . . . . . . . 73
3.6 Kernels, ideals, and quotient rings . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.6.1 Kernels of ring homomorphisms . . . . . . . . . . . . . . . . . . . . . . 74
3.6.2 Ideals of a ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.6.3 Quotient rings, R/I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.6.4 Prime and maximal ideals . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.7 Krull's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.8 UFDs, PIDs, and EDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.8.1 Divisibility in an integral domain . . . . . . . . . . . . . . . . . . . . . 80
3.8.2 Unique factorization domains . . . . . . . . . . . . . . . . . . . . . . . 81
3.8.3 Principal ideal domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.8.4 Euclidean domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
CONTENTS
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3.9 Real and complex polynomial rings R[x] and C[x] . . . . . . . . . . . . . . . . 87 3.9.1 C[x] and the Fundamental Theorem of Algebra . . . . . . . . . . . . . 87 3.9.2 The polynomial ring R[x] . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.10 Rational and integer polynomial rings . . . . . . . . . . . . . . . . . . . . . . . 90 3.10.1 Roots of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.10.2 Gauss's lemma and Eisenstein's criterion . . . . . . . . . . . . . . . . . 92 3.10.3 Prime cyclotomic polynomials . . . . . . . . . . . . . . . . . . . . . . . 95 3.10.4 Polynomial rings with coefficients in a UFD, and polynomial rings in several variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.11 Number fields and their rings of integers . . . . . . . . . . . . . . . . . . . . . 97
4 Groups
99
4.1 Groups and subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.1.1 Definition and basic properties of groups . . . . . . . . . . . . . . . . . 99
4.1.2 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.1.3 Cyclic groups and subgroups . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1.4 Products of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.1.5 Cosets and Lagrange's theorem . . . . . . . . . . . . . . . . . . . . . . 103
4.2 Symmetric Groups Sn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.2.1 Permutations and the symmetric group . . . . . . . . . . . . . . . . . . 104
4.2.2 Even and odd permutations . . . . . . . . . . . . . . . . . . . . . . . . 106
4.2.3 Alternating and dihedral groups . . . . . . . . . . . . . . . . . . . . . . 107
4.3 Cayley's theorem and Cayley graphs . . . . . . . . . . . . . . . . . . . . . . . 110
4.3.1 Cayley's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.3.2 Some small finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.4 The category of groups G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.5 Conjugacy classes and quandles . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.5.1 Conjugacy classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.5.2 Quandles and the operation of conjugation . . . . . . . . . . . . . . . . 117
4.6 Kernels, normal subgroups, and quotient groups . . . . . . . . . . . . . . . . . 120
4.6.1 Kernels of group homomorphisms and normal subgroups . . . . . . . . 120
4.6.2 Quotient groups, and projections : G G/N . . . . . . . . . . . . . 121
4.6.3 Isomorphism theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.6.4 Internal direct products . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.7 Matrix rings and linear groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.7.1 Linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.7.2 The general linear groups GLn(R) . . . . . . . . . . . . . . . . . . . . . 125 4.7.3 Other linear groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.7.4 Projective space and the projective linear group P GLn(F ) . . . . . . . 127 4.8 Structure of finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.8.1 Simple groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.8.2 The Jordan-Ho?lder theorem . . . . . . . . . . . . . . . . . . . . . . . . 131
4.9 Abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.9.1 The category A of Abelian groups . . . . . . . . . . . . . . . . . . . . . 136
4.9.2 Finite Abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
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CONTENTS
Appendices
141
A Background mathematics
143
A.1 Logic and proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A.2 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
A.2.1 Basic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
A.2.2 Functions and relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
A.2.3 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
A.2.4 Axioms of set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
A.3 Ordered structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
A.3.1 Partial orders and posets. . . . . . . . . . . . . . . . . . . . . . . . . . 153
A.3.2 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
A.3.3 Boolean algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
A.4 Axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
A.4.1 Zorn's lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
A.4.2 Well-ordering principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Index
161
List of Figures
1.1 Equilateral triangle with lines of symmetry . . . . . . . . . . . . . . . . . . . . 8 1.2 Unit circle S1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Divisors of 432 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Divisibility up through 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 Cyclic rings Z6, Z19, Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1 Lattice of Gaussian integers Z[i] . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2 Free Boolean ring on two elements . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.3 Lattice of Eisenstein integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.4 Primitive 7th roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.1 Subgroups of S3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.2 Symmetries of a pentagon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.3 Symmetries of a cube and tetrahedron . . . . . . . . . . . . . . . . . . . . . . 109 4.4 Cayley graph for D5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.5 Cayley graph for A4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.6 Distributivity in a involutory quandle . . . . . . . . . . . . . . . . . . . . . . . 118 4.7 A conjugacy class in the quaternion group . . . . . . . . . . . . . . . . . . . . 119 4.8 The conjugacy class of transpositions in S4 . . . . . . . . . . . . . . . . . . . . 120 4.9 The Fano plane Z2P 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.10 The projective plane Z3P 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.11 Cayley graph of the Frobenius group F21 = C7 C3 . . . . . . . . . . . . . . . 134 4.12 Heptahedron on a torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 A.1 Lattice of the Powerset of 4 elements . . . . . . . . . . . . . . . . . . . . . . . 157
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LIST OF FIGURES
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