Modern Algebra: An Introduction, 4th Edition

by John R. Durbin

• ISBN13: 9780471321477
• ISBN10: 0471321478
• Edition: 4th
• Format: Hardcover
• Copyright: 2000-01-01
• Publisher: Wiley

Summary

This book presents an introduction to modern (abstract) algebra covering the basic ideas of groups, rings, and fields. The first part of the book treats ideas that are important but neither abstract nor complicated, and provides practice in handling mathematical statements - their meaning, quantification, negation, and proof. This edition features a new section to give more substance to the introduction to Galois theory, updated lists of references and discussions of topics such as Fermat2s Last Theorem and the finite simple groups.

Table of Contents

Introduction	1	(8)
I. Mappings and Operations	9	(20)
Mappings	9	(6)
Composition. Invertible Mappings	15	(4)
Operations	19	(5)
Composition as an Operation	24	(5)
II. Introduction to Groups	29	(20)
Definition and Examples	29	(4)
Permutations	33	(5)
Subgroups	38	(6)
Groups and Symmetry	44	(5)
III. Equivalence. Congruence. Divisibility	49	(21)
Equivalence Relations	49	(5)
Congruence. The Division Algorithm	54	(4)
Integers Modulo n	58	(4)
Greatest Common Divisors. The Euclidean Algorithm	62	(5)
The Fundamental Theorem of Arithmetic	67	(3)
IV. Groups	70	(30)
Elementary Properties	70	(6)
Generators. Direct Products	76	(4)
Cosets	80	(3)
Lagrange's Theorem	83	(5)
Isomorphism	88	(4)
More on Isomorphism	92	(5)
Cayley's Theorem	97	(3)
V. Group Homomorphisms	100	(14)
Homomorphisms of Groups. Kernels	100	(4)
Quotient Groups	104	(5)
The Fundamental Homomorphism Theorem	109	(5)
VI. Introduction to Rings	114	(18)
Definition and Examples	114	(5)
Integral Domains. Subrings	119	(3)
Fields	122	(4)
Isomorphism. Characteristic	126	(6)
VII. The Familiar Number Systems	132	(23)
Ordered Integral Domains	132	(3)
The Integers	135	(2)
Field of Quotients. The Field of Rational Numbers	137	(4)
Ordered Fields. The Field of Real Numbers	141	(4)
The Field of Complex Numbers	145	(4)
Complex Roots of Unity	149	(6)
VIII. Polynomials	155	(19)
Definition and Elementary Properties	155	(5)
The Division Algorithm	160	(4)
Factorization of Polynomials	164	(4)
Unique Factorization Domains	168	(6)
IX. Quotient Rings	174	(15)
Homomorphism of Rings. Ideals	174	(4)
Quotient Rings	178	(2)
Quotient Rings of F[X]	180	(4)
Factorizations and Ideals	184	(5)
X. Field Extensions	189	(8)
Adjoining Roots	189	(4)
Finite Fields	193	(4)
XI. Polynomial Equations	197	(12)
Roots of a Polynomial	197	(1)
Rational Roots. Conjugate Roots	198	(3)
An Introduction to Galois Theory	201	(5)
Solvability by Radicals	206	(3)
XII. Geometric Constructions	209	(8)
Three Famous Problems	209	(4)
Constructible Numbers	213	(1)
Impossible Constructions	214	(3)
XIII. Application of Permutation Groups	217	(13)
Groups Acting on Sets	217	(4)
Burnside's Counting Theorem	221	(6)
Sylow's Theorem	227	(3)
XIV. Symmetry	230	(25)
Finite Symmetry Groups	230	(7)
Infinite Two-dimensional Symmetry Groups	237	(5)
On Crystallographic Groups	242	(8)
The Euclidean Group	250	(5)
XV. Algebraic Coding	255	(15)
Introduction	255	(3)
Linear Codes	258	(4)
Standard Decoding	262	(3)
Error Probability	265	(5)
XVI. Lattices and Boolean Algebras	270	(20)
Partially Ordered Sets	270	(4)
Lattices	274	(4)
Boolean Algebras	278	(4)
Finite Boolean Algebras	282	(5)
Switching	287	(3)
A. Sets	290	(3)
B. Proofs	293	(5)
C. Mathematical Induction	298	(3)
D. Linear Algebra	301	(5)
E. Solutions to Selected Problems	306	(16)
Photo Credit List	322	(1)
Index	323