1st Edition

A First Graduate Course in Abstract Algebra

By W.J. Wickless Copyright 2004
    252 Pages
    by CRC Press

    250 Pages
    by CRC Press

    Since abstract algebra is so important to the study of advanced mathematics, it is critical that students have a firm grasp of its principles and underlying theories before moving on to further study. To accomplish this, they require a concise, accessible, user-friendly textbook that is both challenging and stimulating. A First Graduate Course in Abstract Algebra is just such a textbook.

    Divided into two sections, this book covers both the standard topics (groups, modules, rings, and vector spaces) associated with abstract algebra and more advanced topics such as Galois fields, noncommutative rings, group extensions, and Abelian groups. The author includes review material where needed instead of in a single chapter, giving convenient access with minimal page turning. He also provides ample examples, exercises, and problem sets to reinforce the material. This book illustrates the theory of finitely generated modules over principal ideal domains, discusses tensor products, and demonstrates the development of determinants. It also covers Sylow theory and Jordan canonical form.

    A First Graduate Course in Abstract Algebra is ideal for a two-semester course, providing enough examples, problems, and exercises for a deep understanding. Each of the final three chapters is logically independent and can be covered in any order, perfect for a customized syllabus.

    Definitions, Examples, Elementary Properties
    Subgroups, Cyclic Groups
    Factorization in Z
    First Problem Set
    Second Problem Set
    Third Problem Set
    Normal Subgroups and Factor Groups
    Fourth Problem Set
    Simple Groups and Composition Series
    Fifth Problem Set
    Symmetric Groups
    Sixth Problem Set
    Conjugacy Classes, p-Groups, Solvable Groups
    Seventh Problem Set
    Direct Products
    Eighth Problem Set
    Sylow Theorems
    Ninth Problem Set
    The Structure of Finite Abelien Groups
    Tenth Problem Set
    Definitions and Elementary Properties
    Eleventh Problem Set
    Homomorphisms, Ideals, and Factors Rings
    Twelfth Problem Set
    Principal Ideal Domains
    Thirteenth Problem Set
    Fourteenth Problem Set
    I[x] is a UFD*
    Fifteenth Problem Set
    Euclidean Domains*
    Sixteenth Problem Set
    Elementary Concepts
    Seventeenth Problem Set
    Free and Projective Modules
    Eighteenth Problem Set
    Tensor Products
    Nineteenth Problem Set
    Finitely Generated Modules Over a PID
    Twentieth Problem Set
    A Structure Theorem
    Twenty-First Problem Set
    Definitions and Glossary
    Time for a Little Set Theory
    A Structure Theorem for Vector Spaces
    Twenty-Second Problem Set
    Finite Remarks on Finite Dimensional Vector Spaces
    Twenty-Third Problem Set
    Matrices and Systems of Equations
    Twenty-Fourth Problem Set
    Linear Transformations and Matrices
    Twenty-Fifth Problem Set
    Twenty-Sixth Problem Set
    Characteristic Values, Vectors, Basis Change
    Twenty-Seventh Problem Set
    Canonical Forms
    Twenty-Eighth Problem Set
    Dual Spaces*
    Twenty-Ninth Problem Set
    Inner Product Spaces*
    Thirtieth Problem Set
    Linear Functionals and Adjoints*
    Thirty-First Problem Set
    Preliminary Results
    Thirty-Second Problem Set
    Straight Edge and Compass Construction
    Thirty-Third Problem Set
    Splitting Fields
    Thirty-Fourth Problem Set
    The Algebraic Closure of a Field*
    Thirty-Fifth Problem Set
    A Structure Theorem for Finite Fields
    Thirty-Sixth Problem Set
    The Galois Correspondence
    Thirty-Seventh Problem Set
    Galois Criterion for Radical Solvability
    Thirty-Eighth Problem Set
    The General Equation of Degree n
    Thirty-Ninth Problem Set
    Simple Models
    Fortieth Problem Set
    The Jacobson Radical
    Forty-First Problem Set
    The Jacobson Density Theorem
    Semisimple Artinian Rings
    Forty-Second Problem Set
    Structure of Complex Group Algebras
    Applications to Finite Groups
    Forty-Third Problem Set
    Exact Sequences and ZG-Modules
    Forty-Fourth Problem Set
    Semidirect Products
    Forty-Fifth Problem Set
    Extensions and Factor Sets
    Forty-Sixth Problem Set
    Solution of the Extension Problem
    Forty-Seventh Problem Set
    Direct Sums and Products
    Forty-Eighth Problem Set
    Structure Theorem for Divisible Groups
    Forty-Ninth Problem Set
    Rank One Torsion-Free Groups
    Fiftieth Problem Set
    Structure of Completely Decomposable Groups
    Fifty-First Problem Set
    Algebraically Compact Groups
    Fifty-Second Problem Set
    Structure of Algebraically Compact Groups
    Fifty-Third Problem Set
    Structure of Countable Torsion Groups
    Fifty-Fourth Problem Set


    Wickless, W.J.

    "This is a very useful text on abstract algebra at the beginning graduate level…the notions of tensor product and projectivity of modules is introduced early and serve in several places to simplify proofs…numerous worked out examples shed light on the abstract theory and help to understand what is going on."
    - Monatshefte für Mathematik