Workbook in Higher Algebra David Surowski Department of Mathematics Kansas State University Manhattan, KS 66506-2602, USA
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Contents Acknowledgement
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1 Group Theory 1.1 Review of Important Basics . . . . . . . 1.2 The Concept of a Group Action . . . . . 1.3 Sylow’s Theorem . . . . . . . . . . . . . 1.4 Examples: The Linear Groups . . . . . . 1.5 Automorphism Groups . . . . . . . . . . 1.6 The Symmetric and Alternating Groups 1.7 The Commutator Subgroup . . . . . . . 1.8 Free Groups; Generators and Relations 2 Field and Galois Theory 2.1 Basics . . . . . . . . . . . . . . . . . . 2.2 Splitting Fields and Algebraic Closure 2.3 Galois Extensions and Galois Groups . 2.4 Separability and the Galois Criterion 2.5 Brief Interlude: the Krull Topology . 2.6 The Fundamental Theorem of Algebra 2.7 The Galois Group of a Polynomial . . 2.8 The Cyclotomic Polynomials . . . . . 2.9 Solvability by Radicals . . . . . . . . . 2.10 The Primitive Element Theorem . . . 3
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1 1 5 12 14 16 22 28 36
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42 42 47 50 55 61 62 62 66 69 70
Elementary Factorization Theory 72 3.1