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Algebra and Galois Theories

Régine Douady and Adrien Douady
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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Most of us were first exposed to Galois theory either as upper-level undergraduates or beginning graduate students, and first learned the material from a strictly algebraic point of view, relating certain kinds of field extensions to groups of automorphisms of these fields.  However, the idea of a “Galois correspondence” extends well beyond algebra; a very similar theory involves topology, specifically regarding covering spaces and fundamental groups. This book explores this topological Galois theory, studying it in parallel with the classical Galois theory. Actually, “parallel” may not be quite the right word: a central theme of the book is that the topological and algebraic theories are not just analogous, but actually impact and enhance each other.   
This book has a long history. It first appeared, in French and in two volumes, back in 1977; much later, in 2005, a new edition, still in French but now in one volume, was published. Probably the most significant textual change in editions was the addition of a new chapter on the theory of dessins d’enfants. The book now under review is an English translation (apparently without revision) of the second French edition. 
Since the publication of the second French edition, a few other books (unseen by me) covering some of the topics contained in this book have appeared. These include Topological Galois Theory by Khovanskii,  Galois Theory, Coverings and Riemann Surfaces by the same author (about 80 pages long!), and Galois Groups and Fundamental Groups by Szamuely.  Nevertheless, books along these lines are still not exactly thick on the ground, so the appearance of this translation certainly adds to the literature.   
The book is made up of seven chapters, each of them divided into sections and subdivided into sub-sections, some but not all of which have titles. The first three chapters (which together made up the original Volume 1) constitute a look at topics in algebra and set theory that most mathematicians have probably encountered before, but perhaps not in such a sophisticated and general way. Chapter 1 discusses set theory via the axiom of choice and Zorn’s Lemma (referred to by that phrase in the text, but in the Introduction as Zorn’s Theorem, probably because in the text it is proved as a consequence of the Axiom of Choice), and also discusses some point-set topology such as Tychonoff’s theorem (proved here using ultrafilters). Chapter 2 looks at category theory, discussing as well such things as profinite groups (to be used later, when discussing infinite-dimensional Galois theory). The third chapter is entitled “Linear Algebra”, but that title fails to do justice to the broad range of topics covered in it. The linear algebra that is covered here is done in the very general context of modules over different kinds of rings, so a great deal of abstract algebra is also covered here.
With this background material covered, the text turns to the meat of the subject. Chapter 4 is topological in nature—covering spaces, the fundamental group, and related issues. As an example of the approach taken here, it should be noted that the fundamental group of a pointed space is defined not in the “usual” way (in terms of homotopic paths) but as the automorphism group of a certain functor.  What I grew up calling the fundamental group, the authors call the Poincaré group; they ultimately prove that for “nice” spaces, they are isomorphic.  
The next chapter is on algebraic Galois theory, but likely not the way you learned it as an undergraduate or early graduate student: it is done here in the more general context of diagonal and etale algebras, and both finite and infinite-dimensional extensions are considered. The approach is very categorical and functorial, emphasizing the fact that the Galois correspondence is an anti-equivalence of categories. 
Having discussed the classical algebraic theory, the authors then turn to the analogous theory for Riemann surfaces. The next chapter, therefore, introduces Riemann algebras and their ramified coverings, and discusses the Galois correspondence in this context. The authors also point out how the algebraic and topological theories each can be used to glean information about the other. In addition, this chapter discusses interesting examples, including the Poincare disc model of hyperbolic geometry.  Homology theory is also developed. 
The final chapter of the book, introduced in the second French edition, discusses the theory of dessin d’enfants, French for “child’s drawing”. For a brief expository introduction to this theory, see the article “What is… A Dessin d’Enfant?” in the August 2003 issue of the Notices of the American Mathematical Society.  Suffice it to say here that these are certain kinds of graphs that can be associated with certain kinds of coverings by Riemann surfaces of the Riemann sphere. Chapter 7 of the text introduces this theory and leads toward a proof of a result called Belyi’s theorem, discussed briefly in the survey article mentioned above. 
This book covers a lot of interesting material and is surely a valuable addition to the literature, but is certainly not for the timid. It brings together a broad array of sophisticated mathematics (abstract algebra, topology, complex analysis, combinatorics, etc.) and it does so in a very general and abstract way, with an exposition that gives whole new meaning to the word “concise”. It is written, I think, very much in the style of Bourbaki, with all the good and bad connotations that that entails. (A friend of mine who happened to look at portions of the book made the prescient comment that it often seemed to “value cleverness over clarity”.) It also assumes a serious mathematical background on the part of the reader; non-measurable sets are mentioned on page 2, for example, and Banach spaces on page 36. 
The language, notation and terminology of the book is occasionally rather idiosyncratic. Most people, for example, would write “20th century” instead of “XX-th century”. The use of Fraktur notation like \( \mathfrak{S} \) and \( \mathfrak{A} \) for the symmetric and alternating groups seems unnecessarily klunky, especially if you’re planning to write this on a blackboard.  What I (and everybody else I know) call the Fundamental Theorem of Algebra is here called “d’Alembert’s theorem”. In addition, as noted earlier, there are a couple of places in the Introduction where the authors speak of “Zorn’s Theorem”.  Vague phrases like “large inequality” (page 2) are used without explanation. The authors refer to “the” choice function defined on the power set of a set, rather than “a” choice function, thereby giving the erroneous impression that there is only one. 
There are also places where things are stated imprecisely enough to be actually false, as, for example, the authors’ statement (page 6) that the product of two ordered sets is, in the lexicographic ordering, always well-ordered. (Presumably the authors meant to say the product of two well-ordered sets is always well-ordered.) Also, the preface contains a tantalizing comment as to how the consideration of topological and algebraic theories in parallel allows for a transcendental proof of “a purely algebraic result without any known purely algebraic proof”, but the result referred to (proposition 6.5.6) does not fit this description at all, and I am still wondering just what result the authors have in mind here. (Perhaps they are referring to the determination, in section 6.4, of the Galois group of the algebraic closure of the field of mermomorphic functions on the Riemann sphere, but I would hardly call that a “purely algebraic” result.)
Like Bourbaki, this text has exercises (which expand the theory considerably, and which are often quite difficult; one, for example, asks the reader to prove the Banach-Tarski paradox, and another calls for a proof of the Hahn-Banach theorem), but, again like Bourbaki, I don’t see it being used as a textbook, except perhaps in very upper-level graduate courses; its “highest and best use”, I think, would be as a reference for mathematicians. There are lots of nice things to be found within its pages, and the general and sophisticated approach is, as noted earlier, not something that is easily found in the literature. 
To summarize and conclude: the likely audience for this book will probably not be very broad, but the people in that audience will probably find this book to be quite valuable, and will be glad that an English translation has, after such a long delay, been produced.
Mark Hunacek ( teaches mathematics at Iowa State University.