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Fundamental Concepts of Abstract Algebra

Gertrude Ehrlich
Publisher: 
Dover Publications
Publication Date: 
2011
Number of Pages: 
340
Format: 
Paperback
Price: 
24.95
ISBN: 
9780486485898
Category: 
Textbook
[Reviewed by
Allen Stenger
, on
09/13/2012
]

This is a competent but uninspiring first course in abstract algebra, concentrating on groups, rings, and fields; but with an extensive coverage of vector spaces, much more than is needed to explain extension fields. It is an unaltered reprint of a 1991 work published by PWS-Kent.

The book includes a large number of exercises, most of moderate difficulty. A novel feature is that each problem section starts with a set of easy true-false questions to test the student’s understanding; for example, an exercise on p. 188 is “the polynomial x4 + 3x3 + 9x2 + 9x + 18 is irreducible over the rational field”, and an exercise on p. 332 is “a regular 680-gon is constructible”.

Another book with similar coverage, but much more interesting and idiosyncratic, is Clark’s Elements of Abstract Algebra.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

  • 1. Preliminaries
    • 1.1 Introduction
    • 1.2 Sets, Relations, and Functions
    • 1.3 The Integers
  • 2. Groups
    • 2.1 Binary Operations
    • 2.2 Groups
    • 2.3 Subgroups, Cyclic Groups, and the Order of an Element
    • 2.4 Isomorphism
    • 2.5 Cosets and Lagrange’s Theorem
    • 2.6 Permutation Groups
    • 2.7 Cayley’s Theorem; Geometric Groups
    • 2.8 Normal Subgroups
    • 2.9 Homomorphism, Factor Groups, and the Fundamental Theorem of Homomorphism for Groups
    • 2.10 Further Isomorphism Theorems; Simple Groups
    • 2.11 Automorphisms and Invariant Subgroups
    • 2.12 Direct Products of Groups
    • 2.13 The Structure of Finite Abelian Groups
    • 2.14 Solvable Groups
    • 2.15 Primary Groups and the Sylow Theorems
  • 3. Rings, Modules, and Vector Spaces
    • 3.1 Rings and Subrings
    • 3.2 Ring Homomorphisms, Ideals, Residue Class Rings, and Simple Rings
    • 3.3 Fundamental Theorem of Homomorphism for Rings
    • 3.4 Maximal and Prime Ideals
    • 3.5 Polynomial Rings
    • 3.6 Principal Ideal Domains
    • 3.7 Euclidean Domains
    • 3.8 Fields of Quotients of Integral Domains
    • 3.9 Polynomials over Unique Factorization Domains
    • 3.10 Groups with Operators; Modules
    • 3.11 Vector Spaces, Subspaces, and Linear Independence
    • 3.12 Basis and Dimension
    • 3.13 Linear Transformations
    • 3.14 Coordinate Vectors, Matrices, and Determinants
    • 3.15 Representation of Linear Transformations by Matrices
    • 3.16 Non-Singular Matrices, Change of Basis, and Similarity
    • 3.17 Eigenvalues and Diagonalization
  • 4. Fields
    • 4.1 Subfields, Extensions, Prime Fields, and Characteristic
    • 4.2 Adjunctions; Algebraic and Transcendental Elements
    • 4.3 Finding an Extension in Which a Given Polynomial Has a Zero
    • 4.4 Algebraic and Transcendental Extensions; Degree of an Extension; Finite Fields
    • 4.5 Classical Constructions I
    • 4.6 Extension of Isomorphisms
    • 4.7 Normal Extensions
    • 4.8 Separable Extensions
    • 4.9 Galois Extensions and Galois Groups
    • 4.10 The Fundamental Theorem of Galois Theory
    • 4.11 Roots of Unity
    • 4.12 Radical Extensions I
    • 4.13 Radical Extensions II
    • 4.14 Transcendence Sets and the Definition of a General Polynomial of Degree n
    • 4.15 Symmetric Functions and the Unsolvability of a General Polynomial of Degree n > 4
    • 4.16 Solution of a General Polynomial of Degree n ≤ 4 by Radicals
    • 4.17 Classical Constructions II; The Fundamental Theorem of Algebra
  • Bibliography
  • Index
Tags: 
Abstract Algebra