×

Galois theory. (English) Zbl 1057.12002

Unquestionably, Galois theory is one of the most fascinating parts of mathematics. Historically, its roots can be traced back to the sixteenth century, when solution formulae for cubic and quartic polynomial equations were first discovered. In the following centuries, the search for both solutions of higher-degree algebraic equations and answers to long-standing, ancient geometric questions culminated in the human drama of Évariste Galois, whose tragic death at age 20 left the mathematical world with his ingenious but not completely developed ideas that eventually led to the mathematical jewel called Galois theory. The full elaboration of Galois’s brilliant approach to understand the roots of polynomials, accomplished during the nineteenth and early twentieth century, was one of the catalysts for the development of modern algebra, including group theory and field theory. Apart from its exciting history, it is the intrinsic beauty, utmost elegance, aesthetic appeal, and deep interaction with many other areas of contemporary mathematics that gives Galois theory its unique richness, fascination, and universal significance. Being of such a kind, Galois theory is virtually typifying the power and beauty of mathematics, the “queen of sciences” (C. F. Gauss), in a very special way, and that is the good reason why the elements of Galois theory are considered to be an indispensable part of the basic knowledge of every serious mathematician. Accordingly, in the current literature, most textbooks of modern abstract algebra contain chapters on Galois theory, varying in their degree of generality, abstraction, depth, and comprehension. Also, there are numerous textbooks devoted exclusively to topics in Galois theory, ranging from elementary introductions to highly advanced treatises, on the one hand, and from historically oriented approaches to systematic-abstract, functorial viewpoints on the other.
The book under review is a highly welcome addition to the already existing abundance of introductions to Galois theory. Like the others, it has been written to serve a particular purpose, thereby justifying its appearance on the scene. First of all, the book is intended for undergraduate students, so that many advanced topics in Galois theory are not covered. On the other hand, the author does discuss a broad range of classical material, and that on more than 500 pages, which makes the book into much more than one of those brief elementary introductions in booklet form, whose ad-hoc style has been designed for very beginners in a one-semester course providing the first steps into the subject. As the author points out in the preface, this book was written in an attempt to do justice to both the history and the power of Galois theory, so that students learn to appreciate the elegance of the theory and, simultaneously, develop a strong sense of where it came from. In this vein, the present new text on Galois theory stands out by its emphatic multi-purpose character, and by the resulting excellent combination of mathematics at its finest, history of modern algebra, culture of mathematical thinking, and instructional mastery. Although the book is comparatively voluminous, it is nowhere long-winded, and the elegance of Galois theory is displayed in its entirety.
As to the precise contents, the text is divided into four parts entitled: I. Polynomials; II. Fields; III. Applications; IV. Further topics. Each part consists of several chapters, and there are 15 chapters all together. The chapters are subdivided into sections which, in turn, consist of subsections, additional mathematical and/or historical notes, and numerous exercises.
Part I comprises the first three chapters and focuses on polynomials. Chapter 1 starts with cubic equations and their solutions, whereas Chapter 2 deals with polynomials of several variables, symmetric polynomials, the fundamental theorem on symmetric functions, and some modern computational aspects with respect to symmetric polynomials. Chapter 3 discusses roots of polynomials, in general, and gives one of the “more algebraic” proofs of the fundamental theorem of algebra.
Part II is formed by Chapters 4 to 7 and here the focus shifts to field extensions and elementary Galois theory. Chapter 4 explains algebraic field extensions. Chapter 5 provides the basics on splitting fields, normal extensions, separable extensions, the theorem of the primitive element, and some related explicit computations. Chapter 6 introduces Galois groups, together with many concrete examples, and Chapter 7 sets the highlight of this part with the description of the Galois correspondence, the so-called fundamental theorem of Galois theory, and first specific applications, including the inverse Galois problem.
Part III (Chapters 8 to 11) treats in great detail the following applications of Galois theory: solvability of algebraic equations by radicals (Chapter 8); cyclotomic extensions (Chapter 9); geometric constructions by ruler and compass (Chapter 10); the structure of finite fields and irreducible polynomials over them (Chapter 11). In particular, the reader will find here such (optional) extras like Origami constructions, Origami numbers, and Berlekamp’s algorithm for computations over finite fields, which is highly enlightening, inspiring, motivating and entertaining. Also, these applications strikingly demonstrate the ubiquity of Galois theory in both mathematics and daily life.
Part IV covers the following further topics, in the context of Galois theory, which essentially serve to round up the overall picture of the story told so far. Chapter 12 analyzes the classical work of Lagrange, Galois, and Kronecker from both the mathematical and historical point of view. Chapter 13 discusses several ideas and methods how to compute Galois groups in concrete cases, whereas Chapter 14 is devoted to the solvability problem for polynomials of prime power degree, together with the allied theory of solvable and primitive permutation groups. The concluding Chapter 15 points to further-going topics (abelian functions and complex multiplication), as the author proves Abel’s theorem for the special case of the lemniscate, thereby introducing all the involved wonderful mathematics that leads the reader right into algebraic and arithmetic geometry.
The text is enhanced by two appendices. Appendix A recalls some prerequisites from abstract algebra, and Appendix B provides hints to selected exercises from the single subsections of the text.
As already mentioned above, the entire text is virtually bursting with carefully selected examples and exercises, additional mathematical and historical remarks, references, hints for further reading, and guiding instructions for practical computations. Everything is adapted, section by section, to the respective topic under discussion, and that provides a great, absolutely unique and invaluable service to the reader.
Evidently, his textbook reflects the beauty and significance of the mathematical jewel “Galois Theory” just as much as the author’s passion, expertise, mathematical culture, and instructional mastery. Despite the limited, principally classical scope of the book, there is a tremendous wealth of knowledge that both students and instructors can gain from this comprehensive, all through delightful text.
One can barely imagine a better introduction to the subject, in all its fascinating aspects, than the one under review, at least not at this level, or a better basis for the study of higher Galois theories (à la A. Grothendieck).

MSC:

12F10 Separable extensions, Galois theory
12-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory
12E12 Equations in general fields
12E20 Finite fields (field-theoretic aspects)
12Y05 Computational aspects of field theory and polynomials (MSC2010)
01A55 History of mathematics in the 19th century
PDFBibTeX XMLCite