Galois Theory Through Exercises

This introductory book attempts to make abstract algebra easier to learn by making it more concrete. It is well-done and I think should be successful in reaching this goal. It is not a problem book, although it consists mostly of problems. All the classical theorems of Galois theory are stated without (immediate) proof, with two or three theorems per chapter, and then there are a large number of exercises to work out particular examples of the theorems, or to apply them to particular problems, or sometimes to prove additional theorems. Most chapters have a “Using Computers” section (based on Maple), but these are usually skimpy and consist of using the computer to find roots or factor polynomials. The main narrative is about 100 pages, then there are 50 pages for the proofs of the theorems that were stated earlier without proof, followed by about 80 pages of hints and solutions to the exercises. There is also a long (50 page) appendix giving background material on groups, rings, and fields.

The narrative is mostly historical. Galois Theory originated in studying the solvability of polynomial equations by radicals, and the book starts with the classical solutions of the quadratic, cubic, and quartic equations. It then spends a good bit of time on field extensions, splitting fields, and field automorphisms; this is all done simply and clearly, and there are lots of specific examples of fields to work on. The treatment is in slightly more generality than needed, as it covers fields of zero and of non-zero characteristic, and it covers separable extensions. There are a few things out of historical order, such as Hilbert’s Theorem 90. Then we get to the main results about solvable groups and solvable equations. It finishes with a chapter on the classical ruler-and-compass construction problems (trisecting the angle, doubling the cube, and squaring the circle), along with Gauss’s work on constructing regular polygons. There’s a final chapter on computing Galois groups, a subject you almost never see in abstract algebra books.

There are a lot of interesting sidelines in the exercises. My favorite (one that I had not seen before) is Nagell’s simple proof (Exercise 13.6) that a quintic with two complex roots and three real roots is not solvable by radicals; this requires a modest knowledge of field extensions and no knowledge of Galois theory. Another gem (actually in the body and not the exercises) is the treatment of the “casus irreducibilis”: mathematicians had long observed that when Cardano’s formula is used to solve a cubic that has three real roots, the formula always generates complex numbers. With Galois Theory we can prove this observation, in the form that the equation is not solvable by real radicals over a real field.

The prerequisites are modest but seem a little vague and perhaps inconsistent. For example, the book carefully defines and explains algebraic and transcendental numbers, but seems to assume that you already know that \(\pi\) is transcendental. The book uses normal subgroups and many parts of group theory without comment, but assumes you have never seen permutation groups and so explains these carefully. The appendix on groups, rings, and fields should fill in any gaps here, but the book does assume you are at least comfortable with these subjects when you begin.

The classic work in this area is Artin’s Galois Theory: Lectures Delivered at the University of Notre Dame, although that takes a completely different approach and probably should not be compared to this one. Another abstract algebra book that is rich in exercises, is Pinter’s A Book of Abstract Algebra; I think its Galois Theory section is especially good. A well-regarded book (that I have not seen) is Cox’s Galois Theory; this seems similar to the present book in many ways but is much longer and more detailed.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.