A Course on Abstract Algebra

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World Scientific, 2010 - Mathematics - 359 pages
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This textbook provides an introduction to abstract algebra for advanced undergraduate students. Based on the author's lecture notes at the Department of Mathematics, National Chung Cheng University of Taiwan, it begins with a description of the algebraic structures of the ring and field of rational numbers. Abstract groups are then introduced. Technical results such as Lagrange's Theorem and Sylow's Theorems follow as applications of group theory. Ring theory forms the second part of abstract algebra, with the ring of polynomials and the matrix ring as basic examples. The general theory of ideals as well as maximal ideals in the rings of polynomials over the rational numbers are also discussed. The final part of the book focuses on field theory, field extensions and then Galois theory to illustrate the correspondence between the Galois groups and field extensions.

This textbook is more accessible and less ambitious than most existing books covering the same subject. Readers will also find the pedagogical material very useful in enhancing the teaching and learning of abstract algebra.

 

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Contents

1 Preliminaries
1
2 Algebraic Structure of Numbers
17
3 Basic Notions of Groups
35
4 Cyclic Groups
53
5 Permutation Groups
65
6 Counting Theorems
81
7 Group Homomorphisms
95
8 The Quotient Group
109
13 Ideals and Quotient Rings
193
14 Ring Homomorphisms
205
15 Polynomial Rings
223
16 Factorization
239
17 Vector Spaces Over an Arbitrary Field
261
18 Field Extensions
273
19 All About Roots
295
20 Galois Pairing
315

9 Finite Abelian Groups
127
10 Sylow Theorems and Applications
143
11 Introduction to Group Presentations
159
12 Types of Rings
177
21 Applications of the Galois Pairing
333
Index
351
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