Fields and Galois theory. (English) Zbl 1103.12001

Howie, John M.

Springer Undergraduate Mathematics Series. London: Springer (ISBN 1-85233-986-1/pbk). x, 225 p. (2006).

The author wrote this book to provide the reader with a treatment of classical Galois theory. He assumes the reader has some knowledge of modern algebra. The book is well written. It contains many examples and over 100 exercises with solutions in the back of the book. Sprinkled throughout the book are interesting commentaries and historical comments. The book is suitable as a textbook for upper level undergraduate or beginning graduate students. Some supplementary material on set theory and group theory may be needed for the student with no background in modern algebra.
The first chapter is devoted to a review of some basic properties of rings, groups, and fields. The second chapter deals with results on Euclidean domains and unique factorization domains that are needed for the study of polynomial rings over a field. Chapter 3 contains basic results from field theory and Chapter 4 deals with applications to geometry such as ruler and compass constructions. After chapters on splitting fields and finite fields, the author presents the results leading up to the fundamental theorem of Galois theory. The next chapter is concerned with quadratic, cubic, and quartic polynomials as they pertain to the solution by radicals. The last three chapters are on groups such as soluble groups, insoluble quintics, and the construction of regular polygons.

Reviewer: John N. Mordeson (Omaha)

MSC:

12F10	Separable extensions, Galois theory
12-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory

Keywords:

groups; rings; fields; Galois theory; ruler and compass constructions; equations; solution by radicals