1st Edition

# A First Graduate Course in Abstract Algebra

252 Pages
by CRC Press

250 Pages
by CRC Press

252 Pages
by CRC Press

Also available as eBook on:

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.

PREFACE
GROUPS (MOSTLY FINITE)
Definitions, Examples, Elementary Properties
Subgroups, Cyclic Groups
Factorization in Z
First Problem Set
Isomorphism
Second Problem Set
Homomorphisms
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
RINGS (MOSTLY DOMAINS)
Definitions and Elementary Properties
Eleventh Problem Set
Homomorphisms, Ideals, and Factors Rings
Twelfth Problem Set
Principal Ideal Domains
Thirteenth Problem Set
Polynomials
Fourteenth Problem Set
I[x] is a UFD*
Fifteenth Problem Set
Euclidean Domains*
Sixteenth Problem Set
MODULES
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
VECTOR SPACES
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
Determinants
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
Thirty-First Problem Set
FIELDS AND GALOIS THEORY
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
Thirty-Eighth Problem Set
The General Equation of Degree n
Thirty-Ninth Problem Set
TOPICS IN NONCOMMUTATIVE RINGS
Introduction
Simple Models
Fortieth Problem Set
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
GROUP EXTENSIONS
Introduction
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
TOPICS IN ABELIAN GROUPS
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
REFERENCES
INDEX

### Biography

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