TAILIEUCHUNG - Workbook in Higher Algebra

But this only touches the surface. Computers are a physical implementation of the rules of (mathematical) computation as described by Alan Turing and others from the mid 1930’s through the early 1940’s. Working with a computer at any level but the most superficial requires that you understand algorithms, how they work, how to show they are correct, and that you are able to construct new algorithms. The only way to get to this point is to study basic algorithms, understand why they work, and even why these algorithms are better (or worse) than others. The highly sophisticated standard algorithms of arithmetic are among the best examples to start. But. | Workbook in Higher Algebra David Surowski Department of Mathematics Kansas State University Manhattan KS 66506-2602 USA dbski@ Contents Acknowledgement iii 1 Group Theory 1 Review of Important Basics. 1 The Concept of a Group Action. 5 Sylow s Theorem. 13 Examples The Linear Groups. 15 Automorphism Groups. 17 The Symmetric and Alternating Groups. 23 The Commutator Subgroup. 29 Free Groups Generators and Relations . 37 2 Field and Galois Theory 43 Basics . 43 Splitting Fields and Algebraic Closure. 48 Galois Extensions and Galois Groups. 51 Separability and the Galois Criterion . 56 Brief Interlude the Krull Topology . 62 The Fundamental Theorem of Algebra . 63 The Galois Group of a Polynomial. 63 The Cyclotomic Polynomials . 67 Solvability by Radicals. 70 The Primitive Element Theorem. 71 3 Elementary Factorization Theory 73 Basics . 73 Unique Factorization Domains . 77 Noetherian Rings and Principal Ideal Domains. 83 i ii CONTENTS Principal Ideal Domains and Euclidean Domains. 86 4 Dedekind Domains 89 A Few Remarks About Module Theory. 89 Algebraic Integer Domains. 93 Oe is a Dedekind Domain. 98 Factorization Theory in Dedekind Domains. 99 The Ideal Class Group of a Dedekind A Characterization of Dedekind Domains .103 5 Module Theory 107 The Basic Homomorphism Direct Products and Sums of Modules over a Principal Ideal Domain .117 Calculation of Invariant Application to a Single Linear Chain Conditions and Series of The Krull-Schmidt Injective and Projective Semisimple Example Group 6 Ring Structure Theory 151 The Jacobson 7 Tensor Products 156 Tensor Product as an Abelian Tensor Product as a Left Tensor Product as an .

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