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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

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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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