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 Abstract Algebra: Theory and Applications is an open-source textbook written by Tom Judson that is designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. Its strengths include a wide range of exercises, both computational and theoretical, plus many nontrivial applications.
The first half of the book presents group theory, through the Sylow theorems, with enough material for a semester-long course. The second-half is suitable for a second semester and presents rings, integral domains, Boolean algebras, vector spaces, and fields, concluding with Galois Theory.
PREFACE
This text is intended for a one- or two-semester undergraduate course in abstract algebra. Traditionally, these courses have covered the theoretical aspects of groups, rings, and fields. However, with the development of computing in the last several decades, applications that involve abstract algebra and discrete mathematics have become increasingly important, and many science, engineering, and computer science students are now electing to minor in mathematics.
Though theory still occupies a central role in the subject of abstract algebra and no student should go through such a course without a good notion of what a proof is, the importance of applications such as coding theory and cryptography has grown significantly.
Until recently most abstract algebra texts included few if any applications. However, one of the major problems in teaching an abstract algebra course is that for many students it is their first encounter with an environment that requires them to do rigorous proofs. Such students often find it hard to see the use of learning to prove theorems and propositions; applied examples help the instructor provide motivation. ...
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Hardcover: 386 pages
Author: Thomas W. Judson
Publisher: Virginia Commonwealth University Mathematics (June 1, 2009)
Language: English
ISBN-10: 0982406223
ISBN-13: 978-0982406229
This text is intended for a one- or two-semester undergraduate course in abstract algebra and covers the traditional theoretical aspects of groups, rings, and fields. Many applications are included, including coding theory and cryptography. The nature of the exercises ranges over several categories; computational, conceptual, and theoretical problems are included.
CONTENTS
Preface vii
0 Preliminaries 1
0.1 A Short Note on Proofs . . . . . . . . . . . . . . . . . . . . . 1
0.2 Sets and Equivalence Relations . . . . . . . . . . . . . . . . . 4
1 The Integers 22
1.1 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . 22
1.2 The Division Algorithm . . . . . . . . . . . . . . . . . . . . . 26
2 Groups 35
2.1 The Integers mod n and Symmetries . . . . . . . . . . . . . . 35
2.2 De nitions and Examples . . . . . . . . . . . . . . . . . . . . 40
2.3 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 Cyclic Groups 56
3.1 Cyclic Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 The Group C . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 The Method of Repeated Squares . . . . . . . . . . . . . . . . 64
4 Permutation Groups 72
4.1 De nitions and Notation . . . . . . . . . . . . . . . . . . . . . 73
4.2 The Dihedral Groups . . . . . . . . . . . . . . . . . . . . . . . 81
5 Cosets and Lagrange's Theorem 89
5.1 Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 Lagrange's Theorem . . . . . . . . . . . . . . . . . . . . . . . 92
5.3 Fermat's and Euler's Theorems . . . . . . . . . . . . . . . . . 94
6 Introduction to Cryptography 97
6.1 Private Key Cryptography . . . . . . . . . . . . . . . . . . . . 98
6.2 Public Key Cryptography . . . . . . . . . . . . . . . . . . . . 101
7 Algebraic Coding Theory 108
7.1 Error-Detecting and Correcting Codes . . . . . . . . . . . . . 108
7.2 Linear Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.3 Parity-Check and Generator Matrices . . . . . . . . . . . . . 121
7.4 Ecient Decoding . . . . . . . . . . . . . . . . . . . . . . . . 128
8 Isomorphisms 138
8.1 De nition and Examples . . . . . . . . . . . . . . . . . . . . . 138
8.2 Direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9 Homomorphisms and Factor Groups 152
9.1 Factor Groups and Normal Subgroups . . . . . . . . . . . . . 152
9.2 Group Homomorphisms . . . . . . . . . . . . . . . . . . . . . 155
9.3 The Isomorphism Theorems . . . . . . . . . . . . . . . . . . . 162
10 Matrix Groups and Symmetry 170
10.1 Matrix Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 170
10.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
11 The Structure of Groups 190
11.1 Finite Abelian Groups . . . . . . . . . . . . . . . . . . . . . . 190
11.2 Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . 195
12 Group Actions 203
12.1 Groups Acting on Sets . . . . . . . . . . . . . . . . . . . . . . 203
12.2 The Class Equation . . . . . . . . . . . . . . . . . . . . . . . 207
12.3 Burnside's Counting Theorem . . . . . . . . . . . . . . . . . . 209
13 The Sylow Theorems 220
13.1 The Sylow Theorems . . . . . . . . . . . . . . . . . . . . . . . 220
13.2 Examples and Applications . . . . . . . . . . . . . . . . . . . 224
14 Rings 232
14.1 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
14.2 Integral Domains and Fields . . . . . . . . . . . . . . . . . . . 237
14.3 Ring Homomorphisms and Ideals . . . . . . . . . . . . . . . . 239
14.4 Maximal and Prime Ideals . . . . . . . . . . . . . . . . . . . . 243
14.5 An Application to Software Design . . . . . . . . . . . . . . . 246
15 Polynomials 256
15.1 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . 257
15.2 The Division Algorithm . . . . . . . . . . . . . . . . . . . . . 261
15.3 Irreducible Polynomials . . . . . . . . . . . . . . . . . . . . . 265
16 Integral Domains 277
16.1 Fields of Fractions . . . . . . . . . . . . . . . . . . . . . . . . 277
16.2 Factorization in Integral Domains . . . . . . . . . . . . . . . . 281
17 Lattices and Boolean Algebras 294
17.1 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
17.2 Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 299
17.3 The Algebra of Electrical Circuits . . . . . . . . . . . . . . . . 305
18 Vector Spaces 312
18.1 De nitions and Examples . . . . . . . . . . . . . . . . . . . . 312
18.2 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
18.3 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . 315
19 Fields 322
19.1 Extension Fields . . . . . . . . . . . . . . . . . . . . . . . . . 322
19.2 Splitting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 333
19.3 Geometric Constructions . . . . . . . . . . . . . . . . . . . . . 336
20 Finite Fields 346
20.1 Structure of a Finite Field . . . . . . . . . . . . . . . . . . . . 346
20.2 Polynomial Codes . . . . . . . . . . . . . . . . . . . . . . . . 351
21 Galois Theory 364
21.1 Field Automorphisms . . . . . . . . . . . . . . . . . . . . . . 364
21.2 The Fundamental Theorem . . . . . . . . . . . . . . . . . . . 370
21.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
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