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Basic Abstract Algebra - P. B. Bhattacharya

Hardcover

Published: 25th November 1994
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This 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.

"...a thorough and surprisingly clean-cut survey of the group/ring/field troika which manages to convey the idea of algebra as a unified enterprise." Ian Stewart, New Scientist

Preface to the second editionp. xiii
Preface to the first editionp. xiv
Glossary of symbolsp. xviii
Preliminaries
Sets and mappingsp. 3
Setsp. 3
Relationsp. 9
Mappingsp. 14
Binary operationsp. 21
Cardinality of a setp. 25
Integers, real numbers, and complex numbersp. 30
Integersp. 30
Rational, real, and complex numbersp. 35
Fieldsp. 36
Matrices and determinantsp. 39
Matricesp. 39
Operations on matricesp. 41
Partitions of a matrixp. 46
The determinant functionp. 47
Properties of the determinant functionp. 49
Expansion of det Ap. 53
Groups
Groupsp. 61
Semigroups and groupsp. 61
Homomorphismsp. 69
Subgroups and cosetsp. 72
Cyclic groupsp. 82
Permutation groupsp. 84
Generators and relationsp. 90
Normal subgroupsp. 91
Normal subgroups and quotient groupsp. 91
Isomorphism theoremsp. 97
Automorphismsp. 104
Conjugacy and G-setsp. 107
Normal seriesp. 120
Normal seriesp. 120
Solvable groupsp. 124
Nilpotent groupsp. 126
Permutation groupsp. 129
Cyclic decompositionp. 129
Alternating group A[subscript n]p. 132
Simplicity of A[subscript n]p. 135
Structure theorems of groupsp. 138
Direct productsp. 138
Finitely generated abelian groupsp. 141
Invariants of a finite abelian groupp. 143
Sylow theoremsp. 146
Groups of orders p[superscript 2], pqp. 152
Rings and modules
Ringsp. 159
Definition and examplesp. 159
Elementary properties of ringsp. 161
Types of ringsp. 163
Subrings and characteristic of a ringp. 168
Additional examples of ringsp. 176
Ideals and homomorphismsp. 179
Idealsp. 179
Homomorphismsp. 187
Sum and direct sum of idealsp. 196
Maximal and prime idealsp. 203
Nilpotent and nil idealsp. 209
Zorn's lemmap. 210
Unique factorization domains and euclidean domainsp. 212
Unique factorization domainsp. 212
Principal ideal domainsp. 216
Euclidean domainsp. 217
Polynomial rings over UFDp. 219
Rings of fractionsp. 224
Rings of fractionsp. 224
Rings with Ore conditionp. 228
Integersp. 233
Peano's axiomsp. 233
Integersp. 240
Modules and vector spacesp. 246
Definition and examplesp. 246
Submodules and direct sumsp. 248
R-homomorphisms and quotient modulesp. 253
Completely reducible modulesp. 260
Free modulesp. 263
Representation of linear mappingsp. 268
Rank of a linear mappingp. 273
Field theory
Algebraic extensions of fieldsp. 281
Irreducible polynomials and Eisenstein criterionp. 281
Adjunction of rootsp. 285
Algebraic extensionsp. 289
Algebraically closed fieldsp. 295
Normal and separable extensionsp. 300
Splitting fieldsp. 300
Normal extensionsp. 304
Multiple rootsp. 307
Finite fieldsp. 310
Separable extensionsp. 316
Galois theoryp. 322
Automorphism groups and fixed fieldsp. 322
Fundamental theorem of Galois theoryp. 330
Fundamental theorem of algebrap. 338
Applications of Galois theory to classical problemsp. 340
Roots of unity and cyclotomic polynomialsp. 340
Cyclic extensionsp. 344
Polynomials solvable by radicalsp. 348
Symmetric functionsp. 355
Ruler and compass constructionsp. 358
Additional topics
Noetherian and artinian modules and ringsp. 367
Hom[subscript R] ([characters not reproducible] M[subscript i], [characters not reproducible] M[subscript i])p. 367
Noetherian and artinian modulesp. 368
Wedderburn-Artin theoremp. 382
Uniform modules, primary modules, and Noether-Lasker theoremp. 388
Smith normal form over a PID and rankp. 392
Preliminariesp. 392
Row module, column module, and rankp. 393
Smith normal formp. 394
Finitely generated modules over a PIDp. 402
Decomposition theoremp. 402
Uniqueness of the decompositionp. 404
Application to finitely generated abelian groupsp. 408
Rational canonical formp. 409
Generalized Jordan form over any fieldp. 418
Tensor productsp. 426
Categories and functorsp. 426
Tensor productsp. 428
Module structure of tensor productp. 431
Tensor product of homomorphismsp. 433
Tensor product of algebrasp. 436
Solutions to odd-numbered problemsp. 438
Selected bibliographyp. 476
Indexp. 477
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521460811
ISBN-10: 0521460816
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 508
Published: 25th November 1994
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 23.39 x 15.6  x 2.87
Weight (kg): 0.89
Edition Number: 2
Edition Type: Revised