Bewersdorff, Jörg

Galois theory for beginners. A historical perspective. Transl. from the German by David Kramer.
[B] Student Mathematical Library 35. Providence, RI: American Mathematical Society (AMS). xx, 180 p. $ 35.00 (2006). ISBN 0-8218-3817-2/pbk

The book under review is a translation of the second edition of the German original (Zbl 0995.12001). First published in 2002 under the title ``Algebra für Einsteiger: Von der Gleichungs\-auflösung zur Galois-Theorie'' (Vieweg Verlag, Wiesbaden, Germany), this elementary introduction to Galois theory saw its second edition only two years later, and its third German edition shortly thereafter. This very fact bespeaks both the friendly acceptance and the undiminished popularity of the author's gentle introduction to Galois theory in Germany, and therefore an English translation to the benefit of a wider international community of readers appears to be appropriate enough.

Certainly, there are numerous excellent textbooks on Galois theory, ranging from elementary primers to highly advanced expositions, but the present one differs from all of them in that the subject is introduced in the most elementary way, with only few prerequisites beyond general college mathematics being needed,thereby making it accessible to average undergraduates, or even to curious high-school students.

Basically, the author's main focus is the history of the classical problem of solving a given polynomial equation in one variable by radicals.

Following the historical path by stressing examples, pioneering ideas, the development of fundamental new concepts and techniques over the centuries, crucial discoveries by great mathematicians, and withal keeping the level of abstraction as generally intelligible as possible, the author has produced both a lovely invitation and a profound first introduction to the realm of Galois theory for everyone.

The algebraic background material is developed as far as needed for the respective topics, and the abstract fundamentals are presented only insofar as they are used in concrete applications. Moreover, abstract proofs, except those in the last chapter on the fundamental theorem of Galois theory, are set off from the main text so that the mathematically unexperienced reader can keep them at a greater distance, without interrupting the reading of the narrative. Of course, such an example driven approach to Galois theory, which favours the historical aspects to the modern conceptual viewpoint, has to rest its argumentation rather on many concrete calculations than on the elegant application of general algebraic principles, but exactly this method serves the particular didactic purpose of the present book.

As to the contents, the ten chapters of the text have been kept unaltered, thus covering the following topics:

1. Cubic equations; 2. Casus irreducibilis: The birth of the complex numbers; 3. Biquadratic equations; 4. Equations of degree $n$ and their properties; 5. The search for additional solution formulas; 6. Equations that can be reduced in degree; 7. The construction of regular polygons; 8. The solution of equations of the fifth degree; 9. The Galois group of an equation; 10. Algebraic structures and Galois theory.

However, the original German edition has been expanded by the addition of a number of exercises following each chapter, which must be seen as a significant enhancement of the text as a whole.

At any rate, this propaedeutical introduction to elementary Galois theory represents a highly valuable contribution to popularizing the subject among students in general. Also, the present book can be seen as a useful complement to the existing standard textbooks in algebra insofar as it provides additional motivation, concrete examples, and historical understanding concerning Galois theory. Likewise, its particular utility for teachers in the field ought to be emphasized.

Finally, it should be mentioned that other historical approaches to elementary Galois theory can be found in the slightly more advanced and comprehensive books by J. P. Escofier [Galois Theory. New York: Springer Verlag (2001; Zbl 0967.12001), Paris: Masson (1997; Zbl 0956.11025)] and by J.-P.Tignol [Galois' Theory of Algebraic Equations. Singapore: World Scientific (2001; Zbl 0972.12001)].

Werner Kleinert (Berlin)

Zentralblatt MATH: Zbl 1114.12002