Basic Abstract AlgebraThis is a self-contained text on abstract algebra for senior undergraduate and senior graduate students, which gives complete and comprehensive coverage of the topics usually taught at this level. The book is divided into five parts. The first part contains fundamental information such as an informal introduction to sets, number systems, matrices, and determinants. The second part deals with groups. The third part treats rings and modules. The fourth part is concerned with field theory. Much of the material in parts II, III, and IV forms the core syllabus of a course in abstract algebra. The fifth part goes on to treat some additional topics not usually taught at the undergraduate level, such as the Wedderburn-Artin theorem for semisimple artinian rings, Noether-Lasker theorem, the Smith-Normal form over a PID, finitely generated modules over a PID and their applications to rational and Jordan canonical forms and the tensor products of modules. Throughout, complete proofs have been given for all theorems without glossing over significant details or leaving important theorems as exercises. In addition, the book contains many examples fully worked out and a variety of problems for practice and challenge. Solution to the odd-numbered problems are provided at the end of the book to encourage the student in problem solving. This new edition contains an introduction to categories and functors, a new chapter on tensor products and a discussion of the new (1993) approach to the celebrated Noether-Lasker theorem. In addition, there are over 150 new problems and examples. |
Contents
Sets and mappings | 3 |
Integers real numbers and complex numbers | 30 |
Matrices and determinants | 39 |
Groups | 61 |
Normal subgroups | 91 |
Normal series | 120 |
Permutation groups | 129 |
Structure theorems of groups | 138 |
Modules and vector spaces | 246 |
Algebraic extensions of fields | 281 |
Normal and separable extensions | 300 |
Galois theory | 322 |
Applications of Galois theory to classical | 340 |
Noetherian and artinian modules and rings | 367 |
Smith normal form over a PID and rank | 392 |
Finitely generated modules over a PID | 402 |
Rings | 159 |
Ideals and homomorphisms | 179 |
Unique factorization domains and euclidean | 212 |
Rings of fractions | 224 |
Integers | 233 |
Tensor products | 426 |
Solutions to oddnumbered problems | 438 |
Selected bibliography | 476 |
Other editions - View all
Common terms and phrases
a₁ abelian group algebraic closure artinian automorphism b₁ bijective binary operation C₁ called Chapter Clearly commutative ring composition series conjugate contains contradiction cosets cyclic group defined Definition degree denote direct sum division ring e₁ example exists extension of F factors field F finite group follows G₁ group G group of order Hence homomorphism idempotent identity implies induction integral domain invertible irreducible polynomial isomorphism left ideal left R-module Lemma Let a,b Let F Let f(x Let G linear m₁ mapping f maximal minimal polynomial module n₁ noetherian nonempty normal extension normal subgroup nth root p₁ permutation polynomial f(x positive integer prime ideal Problems Proof prove R-module R-submodule R₁ right ideal ring with unity roots of f(x Section Show Similarly Solution solvable splitting field subfield subgroup of G submodule subring subset Suppose surjective Sylow p-subgroup symmetries unique vector space x₁ zero