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Foundations of Galois theory. Translated from the 1960 Russian original by Ann Swinfen. With a foreword by P. J. Hilton. Reprint of the 1962 edition. (English) Zbl 1092.12005
Mineola, NY: Dover Publications (ISBN 0-486-43518-0/pbk). x, 109 p. (2004).
The first English edition of M. M. Postnikov’s introductory textbook “Foundations of Galois Theory” appeared in 1962, published by Pergamon Press, London, and The Macmillan Company, New York (1962; Zbl 0098.01601). The book under review is the unabridged republication by Dover Publications, Inc. of this translation from the Russian original (1960; Zbl 0098.01503).
Now as before, this excellent primer and pedagogical masterpiece is an invaluable introduction to the sources and applications of abstract algebra for beginners. Written by a prominent Russian mathematician and famous teacher, who made us the gift of numerous other outstanding textbooks in various fields of mathematics, this booklet offers students a completely self-contained treatment of elementary Galois theory, together with an introduction to the allied theory of groups and fields. It sets itself the modest aim of explaining the basic abstract theory up to the point where it can be applied to prove the unsolvability by radicals of polynomial equations of degree greater than four. Along this pedagogically motivated line, which is very suitable and motivating for beginners, the author masterly avoids any scary sophistication in his exposition. Instead, he offers many clever ad-hoc proofs of the basic theorems, which are perhaps less elegant, but help the reader to dive into mathematical depth with just a minimum of abstract-algebraic background knowledge, thereby motivating her or him for deeper studies in abstract algebra.
As to its methodological mastery, M. M. Postnikov’s introduction to Galois theory can be compared to E. Artin’s classic booklet “Galois Theory” [University of Notre Dame Press (1944; Zbl 0060.04814)]; Dover Publications, Inc. (1998; Zbl 1053.12501)], but it appears to be more elaborated, comprehensive, down-to-earth and beginner-oriented than Artin’s popular standard text. Of course, there is a vast amount of textbooks on Galois theory that have appeared in the meantime, and from which a student or teacher can choose, but M. M. Postnikov’s charming primer will remain a historical milestone in this field of teaching and self-study. Its outstanding value as a first introduction to the subject is undiminished, due to M. M. Postnikov’s inimitable teaching skills and the timeless beauty of Galois theory itself.
For the sake of completeness, here is again a brief description of the contents of this classic textbook: Part I is entitled “The Elements of Galois Theory” and consists of three chapters explaining the basics of field theory, the necessary facts from group theory, and the Galois correspondence, respectively. Part II is devoted to polynomial equations and comes with four chapters, including some higher group theory, especially solvable groups,radical field extensions, normal fields with solvable Galois group, the construction of equations solvable by radicals, and the theorem of the unsolvability by radicals of the general equation of degree greater than four. According to the author’s limited aims, there is no discussion of ruler and compass constructions, which is probably the only little drawback of this otherwise brilliant introduction to Galois theory. Nevertheless, it is very gratifying to see that Postnikov’s classic has been made available again, very much so to the benefit of further generations of students and teachers in the field.

12F10 Separable extensions, Galois theory
12-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory
12E12 Equations in general fields
20-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory
12E10 Special polynomials in general fields