Galois Theory for Beginners is a volume in the Student Mathematical Library series published by the American Mathematical Society. It is a translation of the author’s Algebra für Einsteiger: Gleichungauflösung zur Galois-Theorie (which I translate loosely as “Algebra for Beginners: From the Solution of Equations to Galois Theory”, a title that is perhaps more descriptive). Exercises have also been added to this new edition. The author’s intention is to approach Galois theory in the simplest possible way, and to follow the historical evolution of the ideas.
Most of us who learned Galois theory encountered it after having at least a modest exposure to the theory of groups and fields. In that context, it is not surprising that, in approaching the theory, we were immediately immersed in automorphism groups, field extensions, splitting fields, and all the associated algebraic apparatus. Of course, we knew that the historical motivation came from questions about solutions of polynomial equations, but that often tended to fade into the background.
The author of this book isn’t going to let that happen. The first four chapters of his book have the flavor of the old “theory of equations” that was once (at least, in my father’s time) part of college algebra. The author starts with al-Khwarizmi’s solutions of quadratic equations and moves on to Tartaglia’s methods for solving cubic equations (and Cardano’s largely successful attempt to take credit for Tartaglia’s work). Succeeding chapters take up the birth of complex numbers and Cardano’s work on solution of biquadratic (quartic) equations. The procedures that Cardano published in Ars Magna for solving cubic and biquadratic equations motivated many attempts to find general solutions of fifth degree and higher polynomial equations. These led, at least in part, to a more systematic study of the solution methods that Cardano had described. Viète, in particular, looked at permissible transformations of polynomial equations that do not change the solutions. He also seems to have been the first to find a construction for creating an equation with specified roots despite working with a very cumbersome notation for describing polynomials.
Having gotten us to this point, the author has subtly introduced the symmetric polynomials and the notion that permutations of solutions might be important. Furthermore, he has done this in a very concrete way, building slowly from specific examples. Before he begins with Galois theory proper, the author takes up three additional topics: equations that can be reduced in degree to facilitate solution, special fifth degree equations that are solvable in radicals, and the construction of regular polygons. The latter chapter establishes the connection between constructability and solution of polynomial equations.
By Chapter 9, we are more than ready to see the promised Galois theory. Characteristically, the author begins by concretely with cubic and biquadratic equations, explicitly enumerating permutations that belong to each Galois group. This chapter attempts to follow Galois’ original approach using the so-called Galois resolvent, but without working through all the details.
The last chapter serves as a bridge between the concrete, “elementary” approach of the earlier chapters and the modern point of view. Here the author assumes that the reader has had the equivalent of a semester course in abstract algebra. He presents a fairly standard modern development and proof of the fundamental theorem of Galois theory.
I don’t know that this is the “most elementary way” of approaching Galois theory. Nonetheless, it is possibly the most concrete, moving deliberately from individual examples to the general results. By comparison, Artin’s Galois Theory also takes a direct run at its subject, often using little more than linear algebra, but it does not share the same focus on concrete examples within the context of the historical development.
The exercises in the text are relatively sparse. Supplementary exercises would be needed if this text were to be used for a course. Generally the book is well-written and pleasant to read. There are a few spots where the translation seems a bit awkward, but they are minor and do not affect the readability.
Bill Satzer (email@example.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.