[MATH] Workbook in Higher Algebra

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Contents
Acknowledgement ii
1 Group Theory 1
1.1 Review of Important Basics . . . . . . . . . . . . . . . . . . . 1
1.2 The Concept of a Group Action . . . . . . . . . . . . . . . . . 5
1.3 Sylow’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Examples: The Linear Groups . . . . . . . . . . . . . . . . . . 14
1.5 Automorphism Groups . . . . . . . . . . . . . . . . . . . . . . 16
1.6 The Symmetric and Alternating Groups . . . . . . . . . . . . 22
1.7 The Commutator Subgroup . . . . . . . . . . . . . . . . . . . 28
1.8 Free Groups; Generators and Relations . . . . . . . . . . . . 36
2 Field and Galois Theory 42
2.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2 Splitting Fields and Algebraic Closure . . . . . . . . . . . . . 47
2.3 Galois Extensions and Galois Groups . . . . . . . . . . . . . . 50
2.4 Separability and the Galois Criterion . . . . . . . . . . . . . 55
2.5 Brief Interlude: the Krull Topology . . . . . . . . . . . . . . 61
2.6 The Fundamental Theorem of Algebra . . . . . . . . . . . . 62
2.7 The Galois Group of a Polynomial . . . . . . . . . . . . . . . 62
2.8 The Cyclotomic Polynomials . . . . . . . . . . . . . . . . . . 66
2.9 Solvability by Radicals . . . . . . . . . . . . . . . . . . . . . . 69
2.10 The Primitive Element Theorem . . . . . . . . . . . . . . . . 70
3 Elementary Factorization Theory 72
3.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2 Unique Factorization Domains . . . . . . . . . . . . . . . . . 76
3.3 Noetherian Rings and Principal Ideal Domains . . . . . . . . 81
i
ii CONTENTS
3.4 Principal Ideal Domains and Euclidean Domains . . . . . . . 84
4 Dedekind Domains 87
4.1 A Few Remarks About Module Theory . . . . . . . . . . . . . 87
4.2 Algebraic Integer Domains . . . . . . . . . . . . . . . . . . . . 91
4.3 OE is a Dedekind Domain . . . . . . . . . . . . . . . . . . . . 96
4.4 Factorization Theory in Dedekind Domains . . . . . . . . . . 97
4.5 The Ideal Class Group of a Dedekind Domain . . . . . . . . . 100
4.6 A Characterization of Dedekind Domains . . . . . . . . . . . 101
5 Module Theory 105
5.1 The Basic Homomorphism Theorems . . . . . . . . . . . . . . 105
5.2 Direct Products and Sums of Modules . . . . . . . . . . . . . 107
5.3 Modules over a Principal Ideal Domain . . . . . . . . . . . . 115
5.4 Calculation of Invariant Factors . . . . . . . . . . . . . . . . . 119
5.5 Application to a Single Linear Transformation . . . . . . . . . 123
5.6 Chain Conditions and Series of Modules . . . . . . . . . . . . 129
5.7 The Krull-Schmidt Theorem . . . . . . . . . . . . . . . . . . . 132
5.8 Injective and Projective Modules . . . . . . . . . . . . . . . . 135
5.9 Semisimple Modules . . . . . . . . . . . . . . . . . . . . . . . 142
5.10 Example: Group Algebras . . . . . . . . . . . . . . . . . . . . 146
6 Ring Structure Theory 149
6.1 The Jacobson Radical . . . . . . . . . . . . . . . . . . . . . . 149
7 Tensor Products 154
7.1 Tensor Product as an Abelian Group . . . . . . . . . . . . . . 154
7.2 Tensor Product as a Left S-Module . . . . . . . . . . . . . . . 158
7.3 Tensor Product as an Algebra . . . . . . . . . . . . . . . . . . 163
7.4 Tensor, Symmetric and Exterior Algebra . . . . . . . . . . . . 165
7.5 The Adjointness Relationship . . . . . . . . . . . . . . . . . . 172
A Zorn’s Lemma and some Applications 175

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