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Workbook in Higher Algebra

By Surowski, David

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Book Id: WPLBN0000660734
Format Type: PDF eBook
File Size: 801.85 KB
Reproduction Date: 2005

Title: Workbook in Higher Algebra  
Author: Surowski, David
Volume:
Language: English
Subject: Science., Mathematics, Logic
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Surowski, D. (n.d.). Workbook in Higher Algebra. Retrieved from http://www.netlibrary.net/


Description
Mathematics document containing theorems and formulas.

Excerpt
Excerpt: 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? ...

Table of Contents
Contents 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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