David Surowski - Workbook in Higher Algebra David Surowski...

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Unformatted text preview: Workbook in Higher Algebra David Surowski Department of Mathematics Kansas State University Manhattan, KS 66506-2602, USA dbski@math.ksu.edu Contents Acknowledgement iii 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 . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Examples: The Linear Groups . . . . . . . . . . . . . . . . . . 15 1.5 Automorphism Groups . . . . . . . . . . . . . . . . . . . . . . 17 1.6 The Symmetric and Alternating Groups . . . . . . . . . . . . 23 1.7 The Commutator Subgroup . . . . . . . . . . . . . . . . . . . 29 1.8 Free Groups; Generators and Relations . . . . . . . . . . . . 37 2 Field and Galois Theory 43 2.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.2 Splitting Fields and Algebraic Closure . . . . . . . . . . . . . 48 2.3 Galois Extensions and Galois Groups . . . . . . . . . . . . . . 51 2.4 Separability and the Galois Criterion . . . . . . . . . . . . . 56 2.5 Brief Interlude: the Krull Topology . . . . . . . . . . . . . . 62 2.6 The Fundamental Theorem of Algebra . . . . . . . . . . . . 63 2.7 The Galois Group of a Polynomial . . . . . . . . . . . . . . . 63 2.8 The Cyclotomic Polynomials . . . . . . . . . . . . . . . . . . 67 2.9 Solvability by Radicals . . . . . . . . . . . . . . . . . . . . . . 70 2.10 The Primitive Element Theorem . . . . . . . . . . . . . . . . 71 3 Elementary Factorization Theory 73 3.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.2 Unique Factorization Domains . . . . . . . . . . . . . . . . . 77 3.3 Noetherian Rings and Principal Ideal Domains . . . . . . . . 83 i ii CONTENTS 3.4 Principal Ideal Domains and Euclidean Domains . . . . . . . 86 4 Dedekind Domains 89 4.1 A Few Remarks About Module Theory . . . . . . . . . . . . . 89 4.2 Algebraic Integer Domains . . . . . . . . . . . . . . . . . . . . 93 4.3 O E is a Dedekind Domain . . . . . . . . . . . . . . . . . . . . 98 4.4 Factorization Theory in Dedekind Domains . . . . . . . . . . 99 4.5 The Ideal Class Group of a Dedekind Domain . . . . . . . . . 102 4.6 A Characterization of Dedekind Domains . . . . . . . . . . . 103 5 Module Theory 107 5.1 The Basic Homomorphism Theorems . . . . . . . . . . . . . . 107 5.2 Direct Products and Sums of Modules . . . . . . . . . . . . . 109 5.3 Modules over a Principal Ideal Domain . . . . . . . . . . . . 117 5.4 Calculation of Invariant Factors . . . . . . . . . . . . . . . . . 121 5.5 Application to a Single Linear Transformation . . . . . . . . . 125 5.6 Chain Conditions and Series of Modules . . . . . . . . . . . . 131 5.7 The Krull-Schmidt Theorem . . . . . . . . . . . . . . . . . . . 134 5.8 Injective and Projective Modules . . . . . . . . . . . . . . . . 137 5.9 Semisimple Modules . . . . . . . . . . . . . . . . . . . . . . . 144 5.10 Example: Group Algebras . . . . . . . . . . . . . . . . . . . . 1485....
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David Surowski - Workbook in Higher Algebra David Surowski...

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