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

About the Author

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.

Fernando Q. Gouvêa was born in São Paulo, Brazil and educated at the Universidade de São Paulo and at Harvard University, where he got his Ph.D. with a thesis on p-adic modular forms and Galois representations. He taught at the Universidade de São Paulo (in Brazil) and at Queen’s University (in Canada) before settling at Colby College (in Maine), where he is now the Carter Professor of Mathematics. Gouvêa has written several books: *Arithmetic of p-adic Modular Forms, p-adic Numbers: An Introduction, Arithmetic of Diagonal Hypersurfaces over Finite Fields (with Noriko Yui), Math through the Ages: A Gentle History for Teachers and Others (with William P. Berlinghoff)*, and *Pathways from the Past I and II (also with Berlinghoff)*. Gouvêa was editor of MAA Focus, the newsletter of the Mathematical Association of America, from 1999 to 2010. He is currently editor of MAA Reviews, an online book review service, and of the Carus Mathematical Monographs book series.

### All MAA Guides

1. A Guide to Complex Variables

2. A Guide to Advanced Real Analysis

3. A Guide to Real Variables

4. A Guide to Topology

5. A Guide to Elementary Number Theory

6. A Guide to Advanced Linear Algebra

7. A Guide to Plane Algebraic Curves

8. A Guide to Groups, Rings, and Fields

9. A Guide to Functional Analysis