Ebook: Review of Groups and Galois Theory

This review material is intended for students who are studying Chapters 7 and 8 of the Third Edition of our textbook Abstract Algebra (co-authored with William D. Blair). These chapters correspond roughly to the first semester of the current graduate algebra course at NIU. I have included summaries of most of these sections, together with some general comments. The review problems are intended to have relatively short answers, and to be more typical of exam questions than of standard textbook exercises. Many of them have been used on our comprehensive exam in algebra.

By assuming that this is a review, I have been able make some minor changes in the order of presentation. For example, the first section covers various examples of groups. In presenting these examples, I have introduced some concepts that are not studied until later in the text. I think it is helpful to have the examples collected in one spot, so that you can refer to them as you review.

From the Preface to the Review: "Abstract Algebra begins at the undergraduate level, but Chapters 7 - 9 are written at a level that we consider appropriate for a student who has spent the better part of a year learning abstract algebra. Although it is more sharply focused than the standard graduate level textbooks, and does not go into as much generality, I hope that its features make it a good place to learn about groups and Galois theory, or to review the basic definitions and theorems."