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David Surowski

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

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Workbook in Higher Algebra David Surowski Department of Mathematics Kansas State University Manhattan, KS 66506-2602, USA [email protected]

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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 . . . . . . . . . . . . . . . . . . . . 148 6 Ring Structure Theory 151 6.1 The Jacobson Radical . . . . . . . . . . . . . . . . . . . . . . 151 7 Tensor Products 156 7.1 Tensor Product as an Abelian Group . . . . . . . . . . . . . . 156 7.2 Tensor Product as a Left S -Module . . . . . . . . . . . . . . . 160 7.3 Tensor Product as an Algebra . . . . . . . . . . . . . . . . . . 165 7.4 Tensor, Symmetric and Exterior Algebra . . . . . . . . . . . . 167 7.5 The Adjointness Relationship . . . . . . . . . . . . . . . . . . 175 A Zorn’s Lemma and some Applications 178

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Acknowledgement The present set of notes was developed as a result of Higher Algebra courses that I taught during the academic years 1987-88, 1989-90 and 1991-92. The distinctive feature of these notes is that proofs are not supplied. There are two reasons for this. First, I would hope that the serious student who really intends to master the material will actually try to supply many of the missing proofs. Indeed, I have tried to break down the exposition in such a way that by the time a proof is called for, there is little doubt as to the basic idea of the proof. The real reason, however, for not supplying proofs is that if I have the proofs already in hard copy, then my basic laziness often encourages me not to spend any time in preparing to present the proofs in class. In other words, if I can simply read the proofs to the students, why not? Of course, the main reason for this is obvious; I end up looking like a fool.
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David Surowski - Workbook in Higher Algebra David Surowski...

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