I was very excited to be given the opportunity to review Ayad’s

Galois Theory and Applications, as I am slated to teach Oberlin College’s Galois Theory course during the Fall 2019 semester. What’s more, I’ve had very positive experiences in the past with problems based books like Dixon’s *Problems in Group Theory* and Murty

and Esmonde’s *Problems in Algebraic Number Theory*, and was happy to finally see such a problems based book focused on Galois Theory.

In terms of material, Ayad has covered all of the standard topics. There are sections on polynomials and generalities about fields, algebraic extensions, normal extensions, Galois extensions, finite fields, etc. There are also sections covering a variety of interesting special topics like multivariable polynomials and derivations, not to mention a 130+ page section on algebraic number theory. On the whole the book is suitable for advanced undergraduates and beginning graduate students, though the author’s choice of references (Lang’s *Algebra for Galois Theory* and Marcus’ *Number Fields **for Algebraic Number Theory*) make it fairly clear that this isn’t an appropriate book for those with a weak background in algebra.

In order to discuss the structure of Galois Theory and Applications I find it useful to recall a few quotes from the preface of the first edition of Pólya and Szegö’s excellent two volume Problems and Theorems in Analysis, in which Pólya and Szegö give a thoughtful account of how they believe one learns mathematics and how this informed the writing of their text.

Each volume presents in its first half problems and in its second half their solutions. In the part containing the problems, especially at the beginning of the separate chapters, there are some explanations, which recall the general notions and theorems needed as a background.

One of the things that I like least about the structure of Galois Theory and Applications is that the solution of every exercise appears immediately after the exercise’s statement. While this isn’t necessarily a problem for instructors using the book as a source of homework problems, this unfortunate choice makes the book much less useful for students wishing to learn the material being covered.

Nor do any of the book’s chapters include introductory sections covering the definitions and theorems most relevant to that chapter’s exercises. Although I’ve often ignored these sections when working through problems based books in the past, their absence strips many problems of their proper context and makes the book much harder to be used by students. In some instances, the absence of these sections doesn’t meaningfully detract from the chapter’s usefulness. For instance, even though the statement of Eisenstein’s Irreducibility Criterion is never given in the text, virtually every student working through the book will be aware of it. On the other hand, the fact that Chapter 8 is entitled ”Transcendental extensions, linearly disjoint extensions, Luroth’s Theorem” yet does not contain an easy to find a statement of Luroth’s Theorem seems like a much bigger concern. That the book does not contain an index simply

exacerbates this problem.

A final nit to pick concerning the book’s structure: although the book contains a list of notation used (it appears at the very end of the book), at only two and a half pages long it is woefully incomplete. When I discovered this list it finally became clear to me that as I had guessed, Gal(E/F ) and Gal(E, F ) were both being used to denote the same object. On the other hand, even though the notation Gal(f (x), F ) is used quite a bit in the book, it does not appear at all in the author’s list of notation.

One may group problems from various points of view – according to required previous knowledge, difficulty, method or result. [...] Another section may deal similarly with a theorem which is stated at the beginning (or proved, if this can be done easily and quickly) and is then applied and specialized in several ways.

As I mentioned above, each chapter in *Galois Theory and Applications* is a list of problems followed immediately by their solutions (i.e., Exercise 1, Solution to Exercise 1, Exercise 2, Solution to Exercise 2, etc). That absolutely no exposition occurs between problems gives the chapters a disorienting feel in which it is difficult to discern any sort of structure in the author’s selection of problems or their ordering. I would say that the problems generally get harder throughout a particular chapter, though the increase in difficulty is very far from being monotonic.

This book is no mere collection of problems. Its most important feature is the systematic arrangement of the material which aims to stimulate the reader to independent work and to suggest to him useful lines of thought. We have devoted more time, care and detailed effort to devising the most effective presentation of the material than might be apparent to the uninitiated at first glance.

Unlike Pólya and Szegö’s *Problems and Theorems in Analysis*, *Galois Theory and **Applications* is not meant to serve as a standalone course in Galois Theory. Instead, Ayad’s aim was to create a collection of problems and exercises related to Galois Theory. In this Ayad was certainly successful. *Galois Theory and Applications* contains almost 450 pages of problems and their solutions. These problems range from the routine and concrete to the very abstract. Many are quite challenging. Some of the problems provide accessible presentations of material not normally seen in a first course on Galois Theory. For example, the chapter ”Galois extensions,

Galois groups” begins with a wonderful problem on formally real fields that I plan on assigning to my students this fall. In light of all of this, it is perhaps appropriate that I end this review with a final quote of Pólya and Szegö:

The independent solving of challenging problems will aid the reader far more than that the aphorisms which follow [...]

Benjamin Linowitz

([email protected]) is an Assistant Professor of Mathematics at Oberlin College. His research concerns the theory of arithmetic groups, a fascinating area lying at the intersection of algebraic number theory and differential geometry. He is also interested in the history of mathematics. His website can be found at http://www2.oberlin.edu/faculty/blinowit/.