Elements of Modern Algebra Seventh Edition By Linda Gilbert and Jimmie Gilbert Book

Contents

Preface ……………….. xi
1 Fundamentals ……………….. 1
1.1 Sets ……………….. 1
1.2 Mappings ……………….. 12
1.3 Properties of Composite Mappings (Optional) ……………….. 25
1.4 Binary Operations ……………….. 30
1.5 Permutations and Inverses ……………….. 37
1.6 Matrices ……………….. 42
1.7 Relations ……………….. 55
Key Words and Phrases ……………….. 62
A Pioneer in Mathematics: Arthur Cayley ……………….. 62
2 The Integers ……………….. 65
2.1 Postulates for the Integers (Optional) ……………….. 65
2.2 Mathematical Induction ……………….. 71
2.3 Divisibility ……………….. 81
2.4 Prime Factors and Greatest Common Divisor ……………….. 86
2.5 Congruence of Integers ……………….. 95
2.6 Congruence Classes ……………….. 107
2.7 Introduction to Coding Theory (Optional) ……………….. 114
2.8 Introduction to Cryptography (Optional) ……………….. 123
Key Words and Phrases ……………….. 134
A Pioneer in Mathematics: Blaise Pascal ……………….. 135
3 Groups ……………….. 137
3.1 Definition of a Group ……………….. 137
3.2 Properties of Group Elements ……………….. 145
3.3 Subgroups ……………….. 152
3.4 Cyclic Groups ……………….. 163
3.5 Isomorphisms ……………….. 174
3.6 Homomorphisms ……………….. 183
Key Words and Phrases ……………….. 188
A Pioneer in Mathematics: Niels Henrik Abel ……………….. 189
4 More on Groups ……………….. 191
4.1 Finite Permutation Groups ……………….. 191
4.2 Cayley’s Theorem ……………….. 205
4.3 Permutation Groups in Science and Art (Optional) ……………….. 208
4.4 Cosets of a Subgroup ……………….. 215
4.5 Normal Subgroups ……………….. 223
4.6 Quotient Groups ……………….. 230
4.7 Direct Sums (Optional) ……………….. 239
4.8 Some Results on Finite Abelian Groups (Optional) ……………….. 246
Key Words and Phrases ……………….. 255
A Pioneer in Mathematics: Augustin Louis Cauchy ……………….. 256
5 Rings, Integral Domains, and Fields ……………….. 257
5.1 Definition of a Ring ……………….. 257
5.2 Integral Domains and Fields ……………….. 270
5.3 The Field of Quotients of an Integral Domain ……………….. 276
5.4 Ordered Integral Domains ……………….. 284
Key Words and Phrases ……………….. 291
A Pioneer in Mathematics: Richard Dedekind ……………….. 292
6 More on Rings ……………….. 293
6.1 Ideals and Quotient Rings ……………….. 293
6.2 Ring Homomorphisms ……………….. 303
6.3 The Characteristic of a Ring ……………….. 313
6.4 Maximal Ideals (Optional) ……………….. 319
Key Words and Phrases ……………….. 324
A Pioneer in Mathematics: Amalie Emmy Noether ……………….. 324
7 Real and Complex Numbers ……………….. 325
7.1 The Field of Real Numbers ……………….. 325
7.2 Complex Numbers and Quaternions ……………….. 333
7.3 De Moivre’s Theorem and Roots of Complex Numbers ……………….. 343
Key Words and Phrases ……………….. 352
A Pioneer in Mathematics: William Rowan Hamilton ……………….. 353
8 Polynomials ……………….. 355
8.1 Polynomials over a Ring ……………….. 355
8.2 Divisibility and Greatest Common Divisor ……………….. 367
8.3 Factorization in F[x] ……………….. 375
8.4 Zeros of a Polynomial ……………….. 384
8.5 Solution of Cubic and Quartic Equations by Formulas (Optional) ……………….. 397
8.6 Algebraic Extensions of a Field ……………….. 409
Key Words and Phrases ……………….. 421
A Pioneer in Mathematics: Carl Friedrich Gauss ……………….. 422
A P P E N D I X: The Basics of Logic ……………….. 423
Answers to True/False and Selected
Computational Exercises ……………….. 435
Bibliography ……………….. 499
Index ……………….. 503