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# Elements Of Modern Algebra

**by**Gilbert, Linda

7th

### 9780495561361

0495561363

Hardcover

10/20/2008

Cengage Learning

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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

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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