A first course in abstract algebra / John B. Fraleigh ; historical notes by Victor Katz.

Format:

Book

Author/Creator:

Fraleigh, John B.

Contributor:

Katz, Victor J.

Language:

English

Subjects (All):

Algebra, Abstract.

Physical Description:

xii, 520 pages : illustrations ; 24 cm

Edition:

Seventh edition.

Place of Publication:

Boston : Addison-Wesley, [2003]

Summary:

This is an in-depth introduction to abstract algebra. Focused on groups, rings and fields, it should give students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures. Features include: a classical approach to abstract algebra focussing on applications; an accessible pedagogy including historical notes written by Victor Katz; and a study of group theory.

Contents:

Sets and relations

I. Groups and subgroups. Introduction and examples

Binary operations

Isomorphic binary structures

Groups

Subgroups

Cyclic groups

Generating sets and Cayley digraphs

II. Permutations, cosets, and direct products. Groups of permutations

Orbits, cycles, and the alternating groups

Cosets and the theorem of Lagrange

Direct products and finitely generated Abelian groups

Plane isometries

III. Homomorphisms and factor groups. Homomorphisms

Factor groups

Factor-group computations and simple groups

Group action on a set

Applications of G-sets to counting

IV. Rings and fields. Rings and fields

Integral domains

Fermat's and Euler's theorems

The field of quotients of an integral domain

Rings of polynomials

Factorization of polynomials over a field

Noncommutative examples

Ordered rings and fields

V. Ideals and factor rings. Homomorphisms and factor rings

Prime and maximal ideas

Gröbner bases for ideals

VI. Extension fields. Introduction to extension fields

Vector spaces

Algebraic extensions

Geometric constructions

Finite fields

VII. Advanced group theory. Isomorphism theorems

Series of groups

Sylow theorems

Applications of the Sylow theory

Free Abelian groups

Free groups

Group presentations

VIII. Groups in topology. Simplicial complexes and homology groups

Computations of homology groups

More homology computations and applications

Homological algebra

IX. Factorization. Unique factorization domains

Euclidean domains

Gaussian integers and multiplicative norms

X. Automorphisms and Galois theory. Automorphisms of fields

The isomorphism extension theorem

Splitting fields

Separable extensions

Totally inseparable extensions

Galois theory

Illustrations of Galois theory

Cyclotomic extensions

Insolvability of the quintic

Appendix: Matrix algebra.

Notes:

Includes bibliographical references (pages 483-485) and index.

Other Format:

Online version: Fraleigh, John B. First course in abstract algebra.

ISBN:

0201763907

9780201763904

0321156080

9780321156082

OCLC:

49312505