9780495561361

Elements Of Modern Algebra

by
  • ISBN13:

    9780495561361

  • ISBN10:

    0495561363

  • Edition: 7th
  • Format: Hardcover
  • Copyright: 10/20/2008
  • Publisher: Brooks Cole
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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

Prefacep. xi
Fundamentalsp. 1
Setsp. 1
Mappingsp. 12
Properties of Composite Mappings (Optional)p. 25
Binary Operationsp. 30
Permutations and Inversesp. 37
Matricesp. 42
Relationsp. 55
Key Words and Phrasesp. 62
A Pioneer in Mathematics: Arthur Cayleyp. 62
The Integersp. 65
Postulates for the Integers (Optional)p. 65
Mathematical Inductionp. 71
Divisibilityp. 81
Prime Factors and Greatest Common Divisorp. 86
Congruence of Integersp. 95
Congruence Classesp. 107
Introduction to Coding Theory (Optional)p. 114
Introduction to Cryptography (Optional)p. 123
Key Words and Phrasesp. 134
A Pioneer in Mathematics: Blaise Pascalp. 135
Groupsp. 137
Definition of a Groupp. 137
Properties of Group Elementsp. 145
Subgroupsp. 152
Cyclic Groupsp. 163
Isomorphismsp. 174
Homomorphismsp. 183
Key Words and Phrasesp. 188
A Pioneer in Mathematics: Niels Henrik Abelp. 189
More on Groupsp. 191
Finite Permutation Groupsp. 191
Cayley's Theoremp. 205
Permutation Groups in Science and Art (Optional)p. 208
Cosets of a Subgroupp. 215
Normal Subgroupsp. 223
Quotient Groupsp. 230
Direct Sums (Optional)p. 239
Some Results on Finite Abelian Groups (Optional)p. 246
Key Words and Phrasesp. 255
A Pioneer in Mathematics: Augustin Louis Cauchyp. 256
Rings, Integral Domains, and Fieldsp. 257
Definition of a Ringp. 257
Integral Domains and Fieldsp. 270
The Field of Quotients of an Integral Domainp. 276
Ordered Integral Domainsp. 284
Key Words and Phrasesp. 291
A Pioneer in Mathematics: Richard Dedekindp. 292
More on Ringsp. 293
Ideals and Quotient Ringsp. 293
Ring Homomorphismsp. 303
The Characteristic of a Ringp. 313
Maximal Ideals (Optional)p. 319
Key Words and Phrasesp. 324
A Pioneer in Mathematics: Amalie Emmy Noetherp. 324
Real and Complex Numbersp. 325
The Field of Real Numbersp. 325
Complex Numbers and Quaternionsp. 333
De Moivre's Theorem and Roots of Complex Numbersp. 343
Key Words and Phrasesp. 352
A Pioneer in Mathematics: William Rowan Hamiltonp. 353
Polynomialsp. 355
Polynomials over a Ringp. 355
Divisibility and Greatest Common Divisorp. 367
Factorization in F[x]p. 375
Zeros of a Polynomialp. 384
Solution of Cubic and Quartic Equations by Formulas (Optional)p. 397
Algebraic Extensions of a Fieldp. 409
Key Words and Phrasesp. 421
A Pioneer in Mathematics: Carl Friedrich Gaussp. 422
The Basics of Logicp. 423
Answers to True/False and Selected Computational Exercisesp. 435
Bibliographyp. 499
Indexp. 503
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