Modern Algebra: An Introduction

ISBN-13: 9780470384435 / Angielski / Twarda / 2008 / 352 str.

John R. Durbin

Opis

Engineers and computer scientists who need a basic understanding of algebra will benefit from this accessible book. The sixth edition includes many carefully worked examples and proofs to guide them through abstract algebra successfully. It introduces the most important kinds of algebraic structures, and helps them improve their ability to understand and work with abstract ideas. New and revised exercise sets are integrated throughout the first four chapters. A more in-depth discussion is also included on Galois Theory. The first six chapters provide engineers and computer scientists with the core of the subject and then the book explores the concepts in more detail.

Spis treści

Introduction
I.  Mappings and Operations
1 Mappings
2 Composition
3Operations
4 Composition as an Operation

II.  Introduction to Groups
5 Definition and Examples
6 Permutations
7 Subgroups
8 Groups and Symmetry

III.  Equivalence. Congruence. Divisibility
9 Equivalence Relations
10 Congruence.  The Division Algorithm
11 Integers Modulo n
12 Greatest Common Divisors.  The Euclidean Algorithm
13 Factorization.  Euler′s Phi–Function

IV.  Groups
14 Elementary Properties
15  Generators.  Direct Products
16 Cosets
17 Lagrange′s Theorem.  Cyclic Groups
18 Isomorphism
19 More on Isomorphism
20 Cayley′s Theorem
Appendix: RSA Algorithm

V.  Group Homomorphisms
21 Homomorphisms of Groups.  Kernels
22 Quotient Groups
23 The Fundamental Homomorphism Theorm

VI.  Introduction to Rings
24 Definition and Examples
25 Integral Domains.  Subrings
26 Fields
27 Isomorphism.  Characteristic

VII.  The Familiar Number Systems
28 Ordered Integral Domains
29 The Integers
30 Field of Quotients.  The Field of Rational Numbers
31 Ordered Fields.  The Field of Real Numbers
32 The Field of Complex Numbers
33 Complex Roots of Unity

VIII.  Polynomials
34 Definition and Elementary Properties
35 The Division Algorithm
36 Factorization of Polynomials
37 Unique Factorization Domains

IX.  Quotient Rings
38 Homomorphisms of Rings. Ideals
39 Quotient Rings
40 Quotient Rings of F[X]
41 Factorization and Ideals

X.  Galois Theory: Overview
42 Simple Extensions.  Degree
43 Roots of Polynomials
44 Fundamental Theorem: Introduction

XI.  Galois Theory
45 Galois Groups
46 Splitting Fields.  Galois Groups
47 Separability and Normality
48 Fundamental Theorem of Galois Theory
49 Solvability by Radicals
50 Finite Fields

XII.  Geometric Constructions
51 Three Famous Problems
52 Constructible Numbers
53 Impossible Constructions

XIII.  Solvable and Alternating Groups
54 Isomorphis Theorems.  Solvable Groups
55 Alternation Groups

XIV.  Applications of Permutation Groups
56 Groups Acting on Sets
57 Burnside′s Counting Theorem
58 Sylow′s Theorem

XV.  Symmetry
59 Finite Symmetry Groups
60 Infinite Two–dimensional Symmetry Groups
61 On Crystallographic Groups
62 The Euclidean Group

XVI.  Lattices and Boolean Algebras
63 Partially Ordered Sets
64 Lattices
65 Boolean Algebras
66 Finite Boolean Algebras

A.  Sets
B.  Proofs
C.  Mathematical Induction
D.  Linear Algebra
E.  Solutions to Selected Problems
Photo Credit List
Index of Notation
Index