# A Guide to Groups, Rings, and Fields

### By Fernando Gouvêa

Catalog Code: DOL-48
Print ISBN: 978-0-88385-355-9
Electronic ISBN: 978-1-61444-211-0
325 pp., Hardbound, 2012
List Price: $49.95 Member Price:$39.95
Series: Dolciani Mathematical Expositions

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.

A short list of errata can be found here.

Preface
A Guide to this Guide
1. Algebra: Classical, Modern, and Ultramodern
2. Categories
3. Algebraic Structures
4. Groups and their Representations
5. Rings and Models
6. Fields and Skew Fields
Bibliography
Index of Notations
Index

### Excerpt: Categories (p. 9)

From the standpoint of category theory, all of mathematics is about objects and arrows: groups and homomorphisms, topological spaces and continuous functions, differentiable manifolds and smooth maps, etc. This gives a useful way of thinking about various mathematical theories, but more importantly it highlights connections between different theories, such as going from a topological space to its first homology group. Since categories are about objects and arrows, one expects functors to map objects to objects and arrows to arrows. It is the latter which turns out to be the fundamental insight: “functorial” constructions are important.

For our purposes, category theory is simply a convenient language in which to express relationships between algebraic structures, so we will not explore it in any sort of detail. This does not mean, however, that the theory is only a language. There are indeed theorems, some of them quite important, but here we will content ourselves with a minimal sketch.