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Algebra and Galois theories. 2ème éd., revue et augmentée. (Algèbre et théories galoisiennes.) (French) Zbl 1076.12004
Nouvelle Bibliothèque Mathématique 4. Paris: Cassini (ISBN 2-84225-005-2). 448 p. (2005).
The first edition of this extraordinary textbook on modern abstract algebra and Galois theories was published more than 25 years ago, back then in two separate volumes (Zbl 0428.12018 and Zbl 0428.30034). While Volume I (Chapters I, II, III) provided, in a dense Bourbakian style, a wealth of basic material from set theory, category theory, and commutative algebra, the subsequent Volume II (Chapters IV, V, VI) was devoted to the authors’ pioneering main goal, namely to develop both algebraic Galois theory of field extensions and topological Galois theory of coverings in a parallel fashion, very much so in the spirit of A. Grothendieck’s and N. Bourbaki’s way of thinking in mathematics.
The book under review is the second, revised and enlarged edition of this unique, meanwhile classic text, this time appearing in one single volume. Apart from several re-arrangements, insertion, and methodological improvements, the authors have added a new chapter (Chapter 7) on A. Grothendieck’s theory of “dessins d’enfants” [Esquisse d’une programme, Schneps, Leila (ed.) et al., Geometric Galois actions. 1. Around Grothendieck’s “Esquisse d’un programme”. Proceedings of the conference on geometry and arithmetic of moduli spaces, Luminy, France, August 1995. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 242, 5–48; English translation: 243–283 (1997; Zbl 0901.14001)], together with an up-dating of the relevant bibliography. On the other hand, the main body of the well-established original text has been left entirely intact.
Thus, in this new edition, the material is arranged in seven chapters, each of which is subdivided into up to ten sections.
Chapter 1 provides the basics from set theory and general topology as far as needed in the sequel. This includes the axiom of choice, choice functions and the Hilbert-Bourbaki tau symbol, Zorn’s lemma and its applications, Noetherian orderings, topological ultrafilters, and Tikhonov’s theorem on products of compact topological spaces.
Chapter 2 introduces categories, functors and their morphisms, representable functors, projective and injective limits, adjoint functors, and – with a view toward infinite Galois theory treated in Chapter 5 – pro-finite spaces and profinite groups.
Chapter 3 is devoted to the fundamentals of commutative and (multi-)linear algebra. In the ten sections of this chapter, the authors discuss ring and ideal theory, various classes of rings, the linear algebra of free modules over a ring, the structure of finitely generated modules over a principal ideal domain, Noetherian rings, algebras of polynomials, tensor products of modules and algebras, graded modules, projective and injective modules, complexes, and resolutions of modules.
Chapter 4 turns to the topological aspects pursued in the sequel. Coverings, universal coverings, Galois coverings, the fundamental group, Van Kampen’s theorem, graphs, and loops are the main topics of this chapter, where the interplay between the algebraic and the topological viewpoint is strongly emphasized. For instance, it is shown that the algebraic definition of the fundamental group (as the automorphism group of a certain functor on coverings) coincides with the topological definition of the Poincaré group (via loops) for locally simply connected spaces, which may be regarded as just one typical indication for both the modern style and the rather sophisticated level of the text.
Chapter 5 explains classical algebraic Galois theory in the general categorical context which has been prepared for in the foregoing chapters. Finite algebras over a field, especially diagonal algebras and étale algebras, are here the central objects of study, and the entire framework of classical algebraic Galois theory (finite and infinite) is presented in the language of those algebras and their functorial transformations. At the end, algebraic Galois theory appears as an anti-equivalence of certain categories, thereby revealing its true and very general nature in a striking way.
The analoguous theory for ramified coverings of Riemann surfaces is developed in Chapter 6. This chapter begins with generalities on Riemann surfaces and their ramified coverings, and turns then to the study of finite ramified analytic coverings by means of étale algebras. The analogy with algebraic Galois theory is established by the fundamental theorem stating that there is an anti-equivalence between the category of finite ramified analytic coverings of a connected compact Riemann surface $$B$$ on the one hand, and the category of étale algebras over the meromorphic function field of $$B$$ on the other. In the sequel, the authors demonstrate the power of this analogy by showing how the two Galois theories, the algebraic and the analytic-topological one, indeed clarify and enhance each other, be it by determining certain algebraic Galois groups via transcendental methods, or by investigating special automorphism groups of Riemann surfaces algebraically. In addition, this chapter also discusses triangulations of Riemann surfaces, their simplicial homology, the Riemann-Hurwitz formula, some uniformization theory, the hyperbolic geometry (or Poincaré geometry) of the plane, and pavements of the disk.
The new Chapter 7 enriches the original text by providing an introduction to G. V. Belyĭ’s theorem [On Galois extensions of a maximal cyclotomic field, Math. USSR, Izv. 14, 247–256 (1980); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43, 267–276 (1979; Zbl 0409.12012)] and to A. Grothendieck’s program “dessins d’enfants” [J. Oesterlé, “Dessins d’enfants”, Bourbaki Seminar, Volume 2001/2002, Exposés 894–908, Astérisque 290, 285–305 (2003; Zbl 1076.14040)]. Belyĭ’ s theorem states that a compact Riemann surface is arithmetic, i.e. definable over an algebraic number field, if and only if it can be obtained as a covering of the Riemann sphere ramified only over the locus $$\{0,1,\infty\}$$, and as such it is the starting point of Grothendieck’s program, which amounts to translate, via the theory of ramified coverings of Riemann surfaces, the problem of classifying algebraic number fields into purely combinatorial problems visualized by simple drawings. In the vein of the previous chapters of the book, this approach is used to describe equivalences between various categories which, a priori, comprise objects of totally different nature: algebraic, topological, complex-analytic, and combinatorial. Finally, the authors illustrate the advanced theoretical content of this additional chapter by two instructive examples of concrete Belyĭ polynomials and actions of the mysterious profinite group $$\operatorname{Aut}_{\mathbb{Q}}(\mathbb{Q})$$ on certain trees.
Now as before, this outstanding text presents a wealth of material from algebra, topology, complex-analytic geometry, arithmetic, and combinatorics in a unique manner. Each chapter comes with a rich amount of exercises, grouped according to the single sections of the respective chapter, and these exercises also reflect the Bourbakian style of the book. In fact, they are plentiful, far-ranging, theoretically supplementing, and often extremely challenging. Like throughout the whole text, special attention is paid to analytic examples and applications, thereby demonstrating the unity of mathematics also in its abstract categorical setting.
No doubt, this text presents mathematics at its finest. Written in a highly abstract, sophisticated, concise, rigorous and elegant style, this book is a treasury for advanced readers, who will find it extremely enlightening and inspiring. However, it appears to be less suitable as a primer for beginners in the field, who might be overstrained by just these unique features characterizing it. At any rate, this second, revised and significantly enlarged edition, of “Algebra and Galois Theories” by Régine and Adrien Douady is a great source for researchers, teachers, and seasoned graduate students in the field, and a highly valuable complement to the more elementary textbooks on the subject, likewise. It remains to be desired that an English edition of this outstanding book will follow in the not too far future, as this would do justice to its global significance for the mathematical community worldwide.

##### MSC:
 12F10 Separable extensions, Galois theory 00A05 Mathematics in general 30F10 Compact Riemann surfaces and uniformization 12F20 Transcendental field extensions 12-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory 14H45 Special algebraic curves and curves of low genus 14H30 Coverings of curves, fundamental group 14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)