This book is Artin’s take on Galois theory, as presented in lectures in the early 1940s. The present book is a Dover 1988 unaltered reprint of the 1944 second revised edition from the University of Notre Dame Press.

The approach is very abstract and general, and deals with arbitrary groups of automorphisms of arbitrary fields rather than permutations groups of roots adjoined to a field. The technical innovation here is the avoidance of the “primitive element” theorem, that the extension field can be generating by adjoining a single element to the base field. Hans Zassenhaus says in his obituary of Artin: “But this state of affairs did not satisfy Artin. He took offense of the central role played by the theorem of the existence of a primitive element for finite separable extensions. This statement has no direct relation to the object of the theory which is to investigate the group of an equation, but it was needed at the time as a prerequisite for the proof of the main theorem.” Artin’s approach was very influential and is the method used today in most advanced texts (but not elementary ones). It also popularized representation theory and sparked much research in that area.

The book works best as a monograph rather than a text. It is very concise and contains only a few examples and no exercises. The most familiar uses of Galois theory, constructions with ruler and compass and the unsolvability of the quintic, appear only in a tacked-on appendix of applications by Arthur N. Milgram. There’s no index or glossary of terms, so it’s up to the reader to remember what the terminology means.

It’s not a book for beginners. Beginning students today would be better served by by the Galois theory sections of a more comprehensive algebra text, such as Hungerford’s *Abstract Algebra: An Introduction* or Clark’s *Elements of Abstract Algebra*.

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