A First Graduate Course in Abstract Algebra: (Chapman & Hall/CRC Pure and Applied Mathematics)

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.

Table of Contents:
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 Linear Functionals and Adjoints* 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 Galois Criterion for Radical Solvability Thirty-Eighth Problem Set The General Equation of Degree n Thirty-Ninth Problem Set TOPICS IN NONCOMMUTATIVE RINGS Introduction 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 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