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Exploratory Galois Theory

John Swallow
Cambridge University Press
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Álvaro Lozano-Robledo
, on

Exploratory Galois Theory is designed as a first undergraduate course on field and Galois theory, with a course in abstract algebra — groups and rings — as prerequisite. As a first intuitive approach to Galois theory, the book concentrates on the subfields of the complex numbers.

The first half of the book is dedicated to field theory: polynomial rings, roots, ring homomorphisms; algebraic numbers, field extensions, minimial polynomials; simple extensions, etc. The second half is dedicated to Galois theory: normal extensions and splitting fields; the Galois group and the Galois correspondence; resolvents, discriminants and computation of Galois groups. The last chapter of the book is dedicated to classical topics such as an introduction to Kummer theory and cyclic extensions; characteristic p and finite fields; ruler-and-compass constructions and solvability by radicals. Throughout the book, there are numerous sections which explain how to work with the previously described mathematical objects, using software: Maple and Mathematica (the text includes plenty of screenshots in which the reader can see how to type the needed expressions). For example, the reader learns how to define and factor polynomials, approximate complex roots, how to define and work with algebraic number fields, and of course, how to calculate Galois groups of polynomials and resolvents.

The goal of the author is, in his own words, "to develop Galois theory in as accessible a manner as possible for an undergraduate audience". The reviewer thinks that the goal was very nicely accomplished in this book, where a beautiful and comprehensive exposition of the abstract theory is greatly enhanced by the computational aspects with the help of software. The students who belong to the "calculator religion" will enormously benefit from the numerous hands-on examples and being able to work explicitly with fields and groups on the computer, while the more abstract-minded students will enjoy the excellent mathematical writing of the book. However, as mentioned earlier, the text is a basic introduction to Galois and field theory, mostly concentrating on subfields of the complex numbers, so, depending on the audience, the book's scope might be too narrow.

Finally, the reviewer would like to end this note with a personal concern. The undergraduate student (or at least an algebraically oriented student) will have to purchase books which cover groups, rings, fields, Galois theory and so on (other topics, even if not covered in class, might be handy for the student in the future). Should the instructor choose a couple of books which cover (some of) these topics or should the instructor pick a book which contains all of the previous topics (such as Abstract Algebra by Dummit and Foote)?

Álvaro Lozano-Robledo is H. C. Wang Assistant Professor at Cornell University.

1. Preliminaries; 2. Algebraic numbers, field extensions, and minimal polynomials; 3. Working with algebraic numbers, field extensions, and minimal polynomials; 4. Multiply-generated fields; 5. The Galois correspondence; 6. Some classical topics; Historical note.