Algebra

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World Scientific, 1992 - Mathematics - 350 pages
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This book comes from the first part of the lecture notes which the author used for a first-year graduate algebra course. The aim of this book is not only to give the students quick access to the basic knowledge of algebra, either for future advancement in the field of algebra, or for general background information, but also to show that algebra is truly a master key or a “skeleton key” to many mathematical problems. As one knows, the teeth of an ordinary key prevent it from opening all but one door; whereas the skeleton key keeps only the essential parts, allow it to unlock many doors. The author wishes to present this book as an attempt to re-establish the contacts between algebra and other branches of mathematics and sciences. Many examples and exercises are included to illustrate the power of intuitive approaches to algebra.
 

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Contents

Set theory and Number Theory
1
2 Unique Factorization Theorem
6
3 Congruence
12
4 Chinese Remainder Theorem
20
5 Complex Integers
23
6 Real Numbers and padic Numbers
33
Group theory
47
2 The Transformation Groups on Sets
55
3 Linear Transformation and Matrix
181
4 Module and Module over P I D
196
5 Jordan Canonical Form
214
6 Characteristic Polynomial
223
7 Inner Product and Bilinear form
232
8 Spectral Theory
243
Polynomials in One Variable and Field Theory
252
2 Algebraic Extension
257

3 Subgroups
62
4 Normal Subgroups and Inner Automorphisms
73
5 Automorphism Groups
82
6 pGroups and Sylow Theorems
85
7 JordanHolder Theorem
89
8 Symmetric Group Sn
96
Polynomials
102
2 Polynomial Rings and Quotient Fields
108
3 Unique Factorization Theorem for Polynomials
114
4 Symmetric Polynomial Resultant and Discriminant
130
5 Ideals
144
Linear Algebra
160
2 Basis and Dimension
165
3 Algebraic Closure
271
4 Characteristic and Finite Field
274
5 Separable Algebraic Extension
282
6 Galois Theory
291
7 Solve Equation by Radicals
306
8 Field Polynomial and Field Discriminant
321
9 Liiroths Theorem
326
Appendix
332
A2 Peanos Axioms
333
A3 Homological Algebra
337
Index
343
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