Elements Of Modern Algebra

by Gilbert, Jimmie; Gilbert, Linda

• ISBN13: 9780534402648
• ISBN10: 053440264X
• Edition: 6th
• Format: Hardcover
• Copyright: 2004-08-10
• Publisher: Brooks Cole

Summary

Succeed in modern algebra with ELEMENTS OF MODERN ALGEBRA! With a user-friendly format, this mathematics text provides you with the tools you need to get a good grade. Strategy boxes give you guidance and explanations about techniques and enable you to become more proficient at constructing proofs. A summary of key words and phrases at the end of each chapter help you master the material. A reference section, symbolic marginal notes, an appendix, and numerous examples help you develop your problem solving skills.

Table of Contents

Preface	ix
Fundamentals	1	(56)
Sets	2	(10)
Mappings	12	(10)
Properties of Composite Mappings (Optional)	22	(5)
Binary Operations	27	(5)
Permutations and Inverses	32	(5)
Matrices	37	(11)
Relations	48	(9)
Key Words and Phrases	54	(1)
A Pioneer in Mathematics	55	(2)
Arthur Cayley
The Integers	57	(60)
Postulates for the Integers (Optional)	58	(5)
Mathematical Induction	63	(7)
Divisibility	70	(5)
Prime Factors and Greatest Common Divisor	75	(8)
Congruence of Integers	83	(6)
Congruence Classes	89	(7)
Introduction to Coding Theory (Optional)	96	(8)
Introduction to Cryptography (Optional)	104	(13)
Key Words and Phrases	114	(1)
A Pioneer in Mathematics	115	(2)
Blaise Pascal
Groups	117	(46)
Definition of a Group	118	(13)
Subgroups	131	(10)
Cyclic Groups	141	(9)
Isomorphisms	150	(7)
Homomorphisms	157	(6)
Key Words and Phrases	162	(1)
A Pioneer in Mathematics	162	(1)
Niels Henrik Abel
More on Groups	163	(58)
Finite Permutation Groups	164	(12)
Cayley's Theorem	176	(3)
Permutation Groups in Science and Art (Optional)	179	(5)
Normal Subgroups	184	(12)
Quotient Groups	196	(9)
Direct Sums (Optional)	205	(6)
Some Results on Finite Abelian Groups (Optional)	211	(10)
Key Words and Phrases	219	(1)
A Pioneer in Mathematics	220	(1)
Augustin Louis Cauchy
Rings, Integral Domains, and Fields	221	(32)
Definition of a Ring	222	(11)
Integral Domains and Fields	233	(5)
The Field of Quotients of an Integral Domain	238	(7)
Ordered Integral Domains	245	(8)
Key Words and Phrases	252	(1)
A Pioneer in Mathematics	252	(1)
Richard Dedekind
More on Rings	253	(30)
Ideals and Quotient Rings	254	(8)
Ring Homomorphisms	262	(9)
The Characteristic of a Ring	271	(5)
Maximal Ideals (Optional)	276	(7)
Key Words and Phrases	280	(1)
A Pioneer in Mathematics	280	(3)
Amalie Emmy Noether
Real and Complex Numbers	283	(26)
The Field of Real Numbers	284	(7)
Complex Numbers and Quaternions	291	(8)
De Moivre's Theorem and Roots of Complex Numbers	299	(10)
Key Words and Phrases	308	(1)
A Pioneer in Mathematics	308	(1)
William Rowan Hamilton
Polynomials	309	(56)
Polynomials over a Ring	310	(10)
Divisibility and Greatest Common Divisor	320	(7)
Factorization in F[x]	327	(7)
Zeros of a Polynomial	334	(9)
Solutions of Cubic and Quartic Equations by Formulas (Optional)	343	(9)
Algebraic Extensions of a Field	352	(13)
Key Words and Phrases	362	(1)
A Pioneer in Mathematics	363	(2)
Carl Friedrich Gauss
Appendix: The Basics of Logic	365	(11)
Answers to Selected Computational Exercises	376	(47)
Bibliography	423	(3)
Index	426