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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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