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# Modern Algebra: An Introduction, 6th Edition

**by**John R. Durbin (The Univ. of Texas, Austin)

6th

### 9780470384435

0470384433

Hardcover

12/1/2008

Wiley

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## Summary

Engineers and computer scientists who need a basic understanding of algebra will benefit from this accessible book. The sixth edition includes many carefully worked examples and proofs to guide them through abstract algebra successfully. It introduces the most important kinds of algebraic structures, and helps them improve their ability to understand and work with abstract ideas. New and revised exercise sets are integrated throughout the first four chapters. A more in-depth discussion is also included on Galois Theory. The first six chapters provide engineers and computer scientists with the core of the subject and then the book explores the concepts in more detail.

## Table of Contents

Introduction | p. 1 |

Mappings and Operations | p. 9 |

Mappings | p. 9 |

Composition. Invertible Mappings | p. 15 |

Operations | p. 19 |

Composition as an Operation | p. 25 |

Introduction to Groups | p. 30 |

Definition and Examples | p. 30 |

Permutations | p. 34 |

Subgroups | p. 41 |

Groups and Symmetry | p. 47 |

Equivalence. Congruence. Divisibility | p. 52 |

Equivalence Relations | p. 52 |

Congruence. The Division Algorithm | p. 57 |

Integers Modulo n | p. 61 |

Greatest Common Divisors. The Euclidean Algorithm | p. 65 |

Factorization. Euler's Phi-Function | p. 70 |

Groups | p. 75 |

Elementary Properties | p. 75 |

Generators. Direct Products | p. 81 |

Cosets | p. 85 |

Lagrange's Theorem. Cyclic Groups | p. 88 |

Isomorphism | p. 93 |

More on Isomorphism | p. 98 |

Cayley's Theorem | p. 102 |

RSA Algorithm | p. 105 |

Group Homomorphisms | p. 106 |

Homomorphisms of Groups. Kernels | p. 106 |

Quotient Groups | p. 110 |

The Fundamental Homomorphism Theorem | p. 114 |

Introduction to Rings | p. 120 |

Definition and Examples | p. 120 |

Integral Domains. Subrings | p. 125 |

Fields | p. 128 |

Isomorphism. Characteristic | p. 131 |

The Familiar Number Systems | p. 137 |

Ordered Integral Domains | p. 137 |

The Integers | p. 140 |

Field of Quotients. The Field of Rational Numbers | p. 142 |

Ordered Fields. The Field of Real Numbers | p. 146 |

The Field of Complex Numbers | p. 149 |

Complex Roots of Unity | p. 154 |

Polynomials | p. 160 |

Definition and Elementary Properties | p. 160 |

Appendix to Section 34 | p. 162 |

The Division Algorithm | p. 165 |

Factorization of Polynomials | p. 169 |

Unique Factorization Domains | p. 173 |

Quotient Rings | p. 178 |

Homomorphisms of Rings. Ideals | p. 178 |

Quotient Rings | p. 182 |

Quotient Rings of F[X] | p. 184 |

Factorization and Ideals | p. 187 |

Galois Theory: Overview | p. 193 |

Simple Extensions. Degree | p. 194 |

Roots of Polynomials | p. 198 |

Fundamental Theorem: Introduction | p. 203 |

Galois Theory | p. 207 |

Algebraic Extensions | p. 207 |

Splitting Fields. Galois Groups | p. 210 |

Separability and Normality | p. 214 |

Fundamental Theorem of Galois Theory | p. 218 |

Solvability by Radicals | p. 219 |

Finite Fields | p. 223 |

Geometric Constructions | p. 229 |

Three Famous Problems | p. 229 |

Constructible Numbers | p. 233 |

Impossible Constructions | p. 234 |

Solvable and Alternating Groups | p. 237 |

Isomorphism Theorems and Solvable Groups | p. 237 |

Alternating Groups | p. 240 |

Applications of Permutation Groups | p. 243 |

Groups Acting on Sets | p. 243 |

Burnside's Counting Theorem | p. 247 |

Sylow's Theorem | p. 252 |

Symmetry | p. 256 |

Finite Symmetry Groups | p. 256 |

Infinite Two-Dimensional Symmetry Groups | p. 263 |

On Crystallographic Groups | p. 267 |

The Euclidean Group | p. 274 |

Lattices and Boolean Algebras | p. 279 |

Partially Ordered Sets | p. 279 |

Lattices | p. 283 |

Boolean Algebras | p. 287 |

Finite Boolean Algebras | p. 291 |

Sets | p. 296 |

Proofs | p. 299 |

Mathematical Induction | p. 304 |

Linear Algebra | p. 307 |

Solutions to Selected Problems | p. 312 |

Photo Credit List | p. 326 |

Index of Notation | p. 327 |

Index | p. 330 |

Table of Contents provided by Ingram. All Rights Reserved. |