W. Keith Nicholson

Solutions Manual to Accompany Introduction to Abstract Algebra

• Broschiertes Buch

Produktbeschreibung

Praise for the Third Edition

". . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . ."--Zentralblatt MATH

The Fourth Edition of Introduction to Abstract Algebra continues to provide an accessible approach to the basic structures of abstract algebra: groups, rings, and fields. The book's unique presentation helps readers advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined. Readers can immediately begin to perform computations using abstract concepts that are developed in greater detail later in the text.

The Fourth Edition features important concepts as well as specialized topics, including:
The treatment of nilpotent groups, including the Frattini and Fitting subgroups
Symmetric polynomials
The proof of the fundamental theorem of algebra using symmetric polynomials
The proof of Wedderburn's theorem on finite division rings
The proof of the Wedderburn-Artin theorem

Throughout the book, worked examples and real-world problems illustrate concepts and their applications, facilitating a complete understanding for readers regardless of their background in mathematics. A wealth of computational and theoretical exercises, ranging from basic to complex, allows readers to test their comprehension of the material. In addition, detailed historical notes and biographies of mathematicians provide context for and illuminate the discussion of key topics. A solutions manual is also available for readers who would like access to partial solutions to the book's exercises.

Introduction to Abstract Algebra, Fourth Edition is an excellent book for courses on the topic at the upper-undergraduate and beginning-graduate levels. The book also serves as a valuable reference and self-study tool for practitioners in the fields of engineering, computer science, and applied mathematics.

Produktdetails

• Verlag: Wiley & Sons
• 4. Aufl.
• Seitenzahl: 162
• Erscheinungstermin: 1. Mai 2012
• Englisch
• Abmessung: 254mm x 178mm x 10mm
• Gewicht: 270g
• ISBN-13: 9781118288153
• ISBN-10: 1118288157
• Artikelnr.: 34571616

Inhaltsangabe

0 Preliminaries 1 0.1 Proofs
1 0.2 Sets
2 0.3 Mappings
3 0.4 Equivalences
4 1 Integers and Permutations 6 1.1 Induction
6 1.2 Divisors and Prime Factorization
8 1.3 Integers Modulo 1.4 Permutations
13 2 Groups 17 2.1 Binary Operations
17 2.2 Groups
19 2.3 Subgroups
21 2.4 Cyclic Groups and the Order of an Element
24 2.5 Homomorphisms and Isomorphisms
28 2.6 Cosets and Lagrange's Theorem
30 2.7 Groups of Motions and Symmetries
32 2.8 Normal Subgroups
34 2.9 Factor Groups
36 2.10 The Isomorphism Theorem
38 2.11 An Application to Binary Linear Codes
43 3 Rings 47 3.1 Examples and Basic Properties
47 3.2 Integral Domains and Fields
52 3.3 Ideals and Factor Rings
55 3.4 Homomorphisms
59 3.5 Ordered Integral Domains
62 4 Polynomials 64 4.1 Polynomials
64 4.2 Factorization of Polynomials over a Field
67 4.3 Factor Rings of Polynomials over a Field
70 4.4 Partial Fractions
76 4.5 Symmetric Polynomials
76 5 Factorization in Integral Domains 81 5.1 Irreducibles and Unique Factorization
81 5.2 Principal Ideal Domains
84 6 Fields 88 6.1 Vector Spaces
88 6.2 Algebraic Extensions
90 6.3 Splitting Fields
94 6.4 Finite Fields
96 6.5 Geometric Constructions
98 6.7 An Application to Cyclic and BCH Codes
99 7 Modules over Principal Ideal Domains 102 7.1 Modules
102 7.2 Modules over a Principal Ideal Domain
105 8 p-Groups and the Sylow Theorems 8.1 Products and Factors
108 8.2 Cauchy's Theorem
111 8.3 Group Actions
114 8.4 The Sylow Theorems
116 8.5 Semidirect Products
118 8.6 An Application to Combinatorics
119 9 Series of Subgroups 122 9.1 The Jordan-H¿older Theorem
122 9.2 Solvable Groups
124 9.3 Nilpotent Groups
127 10 Galois Theory 130 10.1 Galois Groups and Separability
130 10.2 The Main Theorem of Galois Theory
134 10.3 Insolvability of Polynomials
138 10.4 Cyclotomic Polynomials and Wedderburn's Theorem
140 11 Finiteness Conditions for Rings and Modules 142 11.1 Wedderburn's Theorem
142 11.2 The Wedderburn-Artin Theorem
143 Appendices 147 Appendix A: Complex Numbers
147 Appendix B: Matrix Arithmetic
148 Appendix C: Zorn's Lemma
149