Overview

Algebraic structures have come to be ubiquitous in mathematics, with almost all mathematicians encountering groups, rings, fields or more exotic related objects during the course of their research. This book presents an overview of some of the most important algebraic structures in modern mathematics, with an emphasis on creating a coherent picture of how they all interact. In addition to the standard material on groups, rings, modules, fields and Galois theory, the book includes discussions of other important topics, including linear groups, group representations, Artinian rings, projective, injective and flat modules, Dedekind domains and central simple algebras. All of the important theorems are discussed, typically without proofs, but often with a discussion of the intuitive ideas behind those proofs. This insightful guide is ideal for both graduate students in mathematics who are beginning their studies, and researchers who wish to understand the bigger picture of the algebraic structures they encounter.

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9780883853559

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Reviews

This book is at least the sixth major title by Brazilian-born and Harvard-educated Gouvea (Colby College), long-time Mathematical Association of America publication's editor. Billed as delivering a concise, graduate-level overview of the theory of groups, rings, and fields, it emphasizes the main ideas and how they hang together, as well as the ways these subjects are used in other parts of mathematics. Topics beyond those of the title, often left out of similar surveys, include linear groups; group representations; the structure of Artinian rings; projective, injective, and flat modules; Dedekind domains; and central simple algebras. Astonishingly, many theorems are presented, both bedrock (e.g., Schur's lemma or Hilbert's theorem 90) and capstone (e.g., the 1962 Feit-Thompson theorem or Wedderburn's finite division ring theorem), rarely with more than the barest of hints at the intuitive ideas behind their proofs. the subject matter belongs to modern algebra, but Gouvea's approach exploits what he dubs ultramodern algebra, i.e., category theory and allied perspectives. Overall, an enjoyable volume. - F.E.J. Linton, CHOICE This expository algebra book is highly unusual-but in a very successful manner. - Tom De Medts, Mathematical Reviews This book is one of the latest in the MAA series of guides to different areas of mathematics intended to summarize the material graduate students need to know in preparing for PhD comprehensive exams. This one in particular, is welcome since the area, abstract algebra is a required exam subject at many universities. The organization of the book is straightforward with three brief introductory chapters (totalling 28 pages) giving a bit of history, a very little bit of category theory, and some observations on the organization of the subject. These are followed by three extended chapters on the three topics names in the book's title. For each of these the relevant definitions are stated, the important theorems are given without proofs, and enough connective tissue is added to knit each into a coherent narrative. The writing style is not formal ( cheerful subgroups, why study modules?', working without an A ) which makes for a pleasant read. At over 300 pages this guide is over twice as long as some of the others in the series and yet it covers the area with virtually no extranious material. The group theory goes beyond the Sylow Theorems and Jordan-Holder with sections on permutation and linear groups and a section on finite group representations. The ring theory chapter goes beyond the factorization theorems to include localization, Hom and tensor products and sections on both Noetheriam and Artmian rings. The field theory chapter presents the basics of Galois Theory and includes brief discussions of algorithms for computation and of the inverse Galois problem. It concludes with a section on division algebra. The book covers a lot of ground but it may still not quite cover all that some departments may put on their exam syllabi. In particular algebraic geometers and topologists will wish for a bit more homological algebra (projective and injective modules are covered but not Tor or Ext). By contrast number theorists, operator theorists, and students in most other areas will find this book most satisfactory in all respects. - CMS Notes

... Billed as delivering a concise, graduate-level overview of the theory of groups, rings, and fields, it emphasizes the main ideas and how they hang together, as well as the ways these subjects are used in other parts of mathematics. ... Overall, an enjoyable volume. - F.E.J. Linton, CHOICE This expository algebra book is highly unusual-but in a very successful manner. - Tom De Medts, Mathematical Reviews This book is one of the latest in the MAA series of guides to different areas of mathematics intended to summarize the material graduate students need to know in preparing for PhD comprehensive exams. This one in particular, is welcome since the area, abstract algebra is a required exam subject at many universities. The organization of the book is straightforward with three brief introductory chapters (totalling 28 pages) giving a bit of history, a very little bit of category theory, and some observations on the organization of the subject. ... number theorists, operator theorists, and students in most other areas will find this book most satisfactory in all respects. - CMS Notes

"… Billed as delivering a concise, graduate-level overview of the theory of groups, rings, and fields, it emphasizes the main ideas and how they hang together, as well as the ways these subjects are used in other parts of mathematics. … Overall, an enjoyable volume."" - F.E.J. Linton, CHOICE ""This expository algebra book is highly unusual—but in a very successful manner."" - Tom De Medts, Mathematical Reviews ""This book is one of the latest in the MAA series of guides to different areas of mathematics intended to summarize the material graduate students need to know in preparing for PhD comprehensive exams. This one in particular, is welcome since the area, abstract algebra is a required exam subject at many universities. The organization of the book is straightforward with three brief introductory chapters (totalling 28 pages) giving a bit of history, a very little bit of category theory, and some observations on the organization of the subject. … number theorists, operator theorists, and students in most other areas will find this book most satisfactory in all respects."" - CMS Notes"

Author Information

Fernando Q. Gouvêa is the Carter Professor of Mathematics at Colby College, Maine. He was editor of MAA Focus, the newsletter of the Mathematical Association of America, from 1999 to 2010 and is currently editor of MAA Reviews and the Carus Mathematical Monographs book series. He has written several books, including Math Through the Ages, with William P. Berlinghoff, which was awarded the 2007 Beckenbach Book Prize.

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