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Algebra : An Approach via Module Theory - William A. Adkins

Algebra

An Approach via Module Theory

Hardcover Published: 23rd April 1999
ISBN: 9780387978390
Number Of Pages: 526

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This book is designed as a text for a first-year graduate algebra course. As necessary background we would consider a good undergraduate linear algebra course. An undergraduate abstract algebra course, while helpful, is not necessary (and so an adventurous undergraduate might learn some algebra from this book). Perhaps the principal distinguishing feature of this book is its point of view. Many textbooks tend to be encyclopedic. We have tried to write one that is thematic, with a consistent point of view. The theme, as indicated by our title, is that of modules (though our intention has not been to write a textbook purely on module theory). We begin with some group and ring theory, to set the stage, and then, in the heart of the book, develop module theory. Having developed it, we present some of its applications: canonical forms for linear transformations, bilinear forms, and group representations. Why modules? The answer is that they are a basic unifying concept in mathematics. The reader is probably already familiar with the basic role that vector spaces play in mathematics, and modules are a generaliza­ tion of vector spaces. (To be precise, modules are to rings as vector spaces are to fields.

Preface
Groupsp. 1
Definitions and Examplesp. 1
Subgroups and Cosetsp. 6
Normal Subgroups, Isomorphism Theorems, and Automorphism Groupsp. 15
Permutation Representations and the Sylow Theoremsp. 22
The Symmetric Group and Symmetry Groupsp. 28
Direct and Semidirect Productsp. 34
Groups of Low Orderp. 39
Exercisesp. 45
Ringsp. 49
Definitions and Examplesp. 49
Ideals, Quotient Rings, and Isomorphism Theoremsp. 58
Quotient Fields and Localizationp. 68
Polynomial Ringsp. 72
Principal Ideal Domains and Euclidean Domainsp. 79
Unique Factorization Domainsp. 92
Exercisesp. 98
Modules and Vector Spacesp. 107
Definitions and Examplesp. 107
Submodules and Quotient Modulesp. 112
Direct Sums, Exact Sequences, and Homp. 118
Free Modulesp. 128
Projective Modulesp. 136
Free Modules over a PIDp. 142
Finitely Generated Modules over PIDsp. 156
Complemented Submodulesp. 171
Exercisesp. 174
Linear Algebrap. 182
Matrix Algebrap. 182
Determinants and Linear Equationsp. 194
Matrix Representation of Homomorphismsp. 214
Canonical Form Theoryp. 231
Computational Examplesp. 257
Inner Product Spaces and Normal Linear Transformationsp. 269
Exercisesp. 278
Matrices over PIDsp. 289
Equivalence and Similarityp. 289
Hermite Normal Formp. 296
Smith Normal Formp. 307
Computational Examplesp. 319
A Rank Criterion for Similarityp. 328
Exercisesp. 337
Bilinear and Quadratic Formsp. 341
Dualityp. 341
Bilinear and Sesquilinear Formsp. 350
Quadratic Formsp. 376
Exercisesp. 391
Topics in Module Theoryp. 395
Simple and Semisimple Rings and Modulesp. 395
Multilinear Algebrap. 412
Exercisesp. 434
Group Representationsp. 438
Examples and General Resultsp. 438
Representations of Abelian Groupsp. 451
Decomposition of the Regular Representationp. 453
Charactersp. 462
Induced Representationsp. 479
Permutation Representationsp. 496
Concluding Remarksp. 503
Exercisesp. 505
Appendixp. 507
Bibliographyp. 510
Index of Notationp. 511
Index of Terminologyp. 517
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780387978390
ISBN-10: 0387978399
Series: Graduate Texts in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 526
Published: 23rd April 1999
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 3.18
Weight (kg): 0.93
Edition Number: 2
Edition Type: Revised

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