Elements Of Modern Algebra

by Gilbert, Linda

• ISBN13: 9780495561361
• ISBN10: 0495561363
• Edition: 7th
• Format: Hardcover
• Copyright: 2008-10-20
• Publisher: Brooks Cole

Summary

ELEMENTS OF MODERN ALGEBRA 7e, with its user-friendly format, provides you with the tools you need to get succeed in abstract algebra and develop mathematical maturity as a bridge to higher-level mathematics courses.. 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	p. xi
Fundamentals	p. 1
Sets	p. 1
Mappings	p. 12
Properties of Composite Mappings (Optional)	p. 25
Binary Operations	p. 30
Permutations and Inverses	p. 37
Matrices	p. 42
Relations	p. 55
Key Words and Phrases	p. 62
A Pioneer in Mathematics: Arthur Cayley	p. 62
The Integers	p. 65
Postulates for the Integers (Optional)	p. 65
Mathematical Induction	p. 71
Divisibility	p. 81
Prime Factors and Greatest Common Divisor	p. 86
Congruence of Integers	p. 95
Congruence Classes	p. 107
Introduction to Coding Theory (Optional)	p. 114
Introduction to Cryptography (Optional)	p. 123
Key Words and Phrases	p. 134
A Pioneer in Mathematics: Blaise Pascal	p. 135
Groups	p. 137
Definition of a Group	p. 137
Properties of Group Elements	p. 145
Subgroups	p. 152
Cyclic Groups	p. 163
Isomorphisms	p. 174
Homomorphisms	p. 183
Key Words and Phrases	p. 188
A Pioneer in Mathematics: Niels Henrik Abel	p. 189
More on Groups	p. 191
Finite Permutation Groups	p. 191
Cayley's Theorem	p. 205
Permutation Groups in Science and Art (Optional)	p. 208
Cosets of a Subgroup	p. 215
Normal Subgroups	p. 223
Quotient Groups	p. 230
Direct Sums (Optional)	p. 239
Some Results on Finite Abelian Groups (Optional)	p. 246
Key Words and Phrases	p. 255
A Pioneer in Mathematics: Augustin Louis Cauchy	p. 256
Rings, Integral Domains, and Fields	p. 257
Definition of a Ring	p. 257
Integral Domains and Fields	p. 270
The Field of Quotients of an Integral Domain	p. 276
Ordered Integral Domains	p. 284
Key Words and Phrases	p. 291
A Pioneer in Mathematics: Richard Dedekind	p. 292
More on Rings	p. 293
Ideals and Quotient Rings	p. 293
Ring Homomorphisms	p. 303
The Characteristic of a Ring	p. 313
Maximal Ideals (Optional)	p. 319
Key Words and Phrases	p. 324
A Pioneer in Mathematics: Amalie Emmy Noether	p. 324
Real and Complex Numbers	p. 325
The Field of Real Numbers	p. 325
Complex Numbers and Quaternions	p. 333
De Moivre's Theorem and Roots of Complex Numbers	p. 343
Key Words and Phrases	p. 352
A Pioneer in Mathematics: William Rowan Hamilton	p. 353
Polynomials	p. 355
Polynomials over a Ring	p. 355
Divisibility and Greatest Common Divisor	p. 367
Factorization in F[x]	p. 375
Zeros of a Polynomial	p. 384
Solution of Cubic and Quartic Equations by Formulas (Optional)	p. 397
Algebraic Extensions of a Field	p. 409
Key Words and Phrases	p. 421
A Pioneer in Mathematics: Carl Friedrich Gauss	p. 422
The Basics of Logic	p. 423
Answers to True/False and Selected Computational Exercises	p. 435
Bibliography	p. 499
Index	p. 503
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