A Guide to Groups, Rings, and Fields

Fernando Gouvêa

doi:10.5948/UPO9781614442110

This Guide offers a concise overview of the theory of groups, rings, and fields at the graduate level, emphasizing those aspects that are useful in other parts of mathematics. It focuses on the main ideas and how they hang together. It will be useful to both students and professionals. In addition to the standard material on groups, rings, modules, fields, and Galois theory, the book includes discussions of other important topics that are often omitted in the standard graduate course, including linear groups, group representations, the structure of Artinian rings, projective, injective and flat modules, Dedekind domains, and central simple algebras. All of the important theorems are discussed, without proofs but often with a discussion of the intuitive ideas behind those proofs. Those looking for a way to review and refresh their basic algebra will benefit from reading this Guide, and it will also serve as a ready reference for mathematicians who make use of algebra in their work.

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A Guide to Groups, Rings, and Fields

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Contents

Frontmatter
pp i-vi

Contents
pp vii-xiv

Preface
pp xv-xviii

A Guide to this Guide
pp 1-2

1 - Algebra: Classical, Modern, and Ultramodern
pp 3-8

2 - Categories
pp 9-16

3 - Algebraic Structures
pp 17-28

4 - Groups and their Representations
pp 29-106

5 - Rings and Modules
pp 107-220

6 - Fields and Skew Fields
pp 221-276

Bibliography
pp 277-282

Index of Notations
pp 283-294

Index
pp 295-308

About the Author
pp 309-309

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A Guide to Groups, Rings, and Fields

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