A First Course in Abstract Algebra by John B. Fraleigh

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Book Summary of A First Course in Abstract Algebra

Considered a classic by many, A First Course in Abstract Algebra is an in-depth introduction to abstract algebra. Focused on groups, rings and fields, this text gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures.

Features

• This classical approach to abstract algebra focuses on applications.
• The text is geared toward high-level courses at schools with strong mathematics programs.
• Accessible pedagogy includes historical notes written by Victor Katz, an authority on the history of math.
• By opening with a study of group theory, this text provides students with an easy transition to axiomatic mathematics.

Table Of Contents
GROUPS AND SUBGROUPS.

• Introduction and Examples.
• Binary Operations.
• Isomorphic Binary Structures.
• Groups.
• Subgroups.
• Cyclic Groups.
• Generators and Cayley Digraphs.

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.

HOMOMORPHISMS AND FACTOR GROUPS.

• Homomorphisms.
• Factor Groups.
• Factor-Group Computations and Simple Groups.
• Group Action on a Set.
• Applications of G-Sets to Counting.

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.

IDEALS AND FACTOR RINGS.

• Homomorphisms and Factor Rings.
• Prime and Maximal Ideas.
• Gröbner Bases for Ideals.

EXTENSION FIELDS.

• Introduction to Extension Fields.
• Vector Spaces.
• Algebraic Extensions.
• Geometric Constructions.
• Finite Fields.

ADVANCED GROUP THEORY.

• Isomorphism Theorems.
• Series of Groups.
• Sylow Theorems.
• Applications of the Sylow Theory.
• Free Abelian Groups.
• Free Groups.
• Group Presentations.

GROUPS IN TOPOLOGY.

• Simplicial Complexes and Homology Groups.
• Computations of Homology Groups.
• More Homology Computations and Applications.
• Homological Algebra.

FACTORIZATION.

• Unique Factorization Domains.
• Euclidean Domains.
• Gaussian Integers and Multiplicative Norms.

MORPHISMS 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.

Details of Book: A First Course in Abstract Algebra

Book:	A First Course in Abstract Algebra
Author:	John B. Fraleigh
ISBN:	8177589008
ISBN-13:	9788177589009 ,  978-8177589009
Binding:	Paperback
Publishing Date:	2003
Publisher:	Pearson Education
Edition:	7thEdition
Number of Pages:	530
Language:	English
Dimensions:	9.13 x 7.48 x 0.94 inches
Weight:	821 grams