Elements of Modern Algebra

by Gilbert, Jimmie; Gilbert, Linda

• ISBN13: 9780534373511
• ISBN10: 0534373518
• Format: Hardcover
• Copyright: 1999-09-10
• 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

Fundamentals	1	(52)
Sets	2	(9)
Mappings	11	(11)
Properties of Composite Mappings (Optional)	22	(4)
Binary Operations	26	(10)
Matrices	36	(10)
Relations	46	(7)
Key Words and Phrases	51	(1)
A Pioneer in Mathematics: Arthur Cayley	52	(1)
The Integers	53	(56)
Postulates for the Integers (Optional)	54	(4)
Mathematical Induction	58	(8)
Divisibility	66	(4)
Prime Factors and Greatest Common Divisor	70	(7)
Congruence of Integers	77	(6)
Congruence Classes	83	(7)
Introduction to Coding Theory (Optional)	90	(8)
Introduction to Cryptography (Optional)	98	(11)
Key Words and Phrases	107	(1)
A Pioneer in Mathematics: Blaise Pascal	108	(1)
Groups	109	(44)
Definition of a Group	110	(12)
Subgroups	122	(9)
Cyclic Groups	131	(9)
Isomorphisms	140	(7)
Homomorphisms	147	(6)
Key Words and Phrases	151	(1)
A Pioneer in Mathematics: Niels Henrik Abel	152	(1)
More on Groups	153	(54)
Finite Permutation Groups	154	(11)
Cayley's Theorem	165	(3)
Permutation Groups in Science and Art (Optional)	168	(5)
Normal Subgroups	173	(11)
Quotient Groups	184	(8)
Direct Sums (Optional)	192	(5)
Some Results on Finite Abelian Groups (Optional)	197	(10)
Key Words and Phrases	205	(1)
A Pioneer in Mathematics: Augustin Louis Cauchy	206	(1)
Rings, Integral Domains, and Fields	207	(31)
Definition of a Ring	208	(10)
Integral Domains and Fields	218	(5)
The Field of Quotients of an Integral Domain	223	(7)
Ordered Integral Domains	230	(8)
Key Words and Phrases	236	(1)
A Pioneer in Mathematics: Richard Dedekind	237	(1)
More on Rings	238	(27)
Ideals and Quotient Rings	239	(7)
Ring Homomorphisms	246	(9)
The Characteristic of a Ring	255	(5)
Maximal Ideals (Optional)	260	(5)
Key Words and Phrases	263	(1)
A Pioneer in Mathematics: Amalie Emmy Noether	264	(1)
Real and Complex Numbers	265	(25)
The Field of Real Numbers	266	(7)
Complex Numbers and Quaternions	273	(8)
De Moivre's Theorem and Roots of Complex Numbers	281	(9)
Key Words and Phrases	289	(1)
A Pioneer in Mathematics: William Rowan Hamilton	289	(1)
Polynomials	290	(45)
Polynomials over a Ring	291	(9)
Divisibility and Greatest Common Divisor	300	(7)
Factorization in F[x]	307	(7)
Zeros of a Polynomial	314	(9)
Algebraic Extensions of a Field	323	(12)
Key Words and Phrases	333	(1)
A Pioneer in Mathematics: Carl Friedrich Gauss	334	(1)
Appendix: The Basics of Logic	335	(11)
Bibliography	346	(3)
Answers to Selected Computational Exercises	349	(39)
Index	388