This algebra text grew out of the author’s courses taught at École Polytechnique, Palaiseau, France, some years ago. Geared toward advanced undergraduate students, the book focuses on those parts of abstract algebra which primarily deal with the structure of fields, polynomial equations, Galois theory, and the related algebraic theory of differential equations. Thus, in contrast to most introductory textbooks of modern algebra, the author does not follow the common pattern of discussing as many algebraic structure theories as possible, and that both in a strict systematic order and in their greatest possible generality, but provides a guide to the subject along a very particular, interesting and streamlined path linking some important historical aspects to their corresponding modern conceptual principles, including numerous illustrating examples and applications. In this vein, the central theme of the text is field theory, together with its relations to some other areas in abstract algebra and to analysis. As to the precise contents, the text is organized in six chapters, each of which is divided in several thematic sections.
Chapter 1 begins with the motivating classical problem of geometric constructibility by ruler and compass, before introducing then the basics of field theory, including algebraic field extensions and the discussion of the impossibility of solving some of the famous classical problems of constructibility. This chapter also deals with symmetric functions and, in an appendix, with the proofs of the transcendence of the analytic constants \(e\) and \(\pi\).
Chapter 2 turns to polynomial equations and their possible roots, thereby developing the notions of splitting fields, of algebraically closed fields, and of the algebraic closure of a field. The necessary facts from the theory of commutative rings and their ideals are proven in two appendices, with a special emphasis laid on polynomial rings. As an instructive application, the author also explains both the algebraic and the analytic aspects of Puiseux’s theorem on the parametrization of the roots of an algebraic equation with varying coefficients. Chapter 3 is the heart of the book. Moving quickly to Galois extensions, the author describes the Galois correspondence by using Artin’s method. The necessary prerequisites from the theory of separable field extensions are provided along the way, together with the characterization of Galois groups as permutation groups. This chapter also includes some material on discriminants and resolvent polynomials as well as on finite fields.
Chapter 4 is a digression into group theory and explains the basics of the subject as far as needed in Galois theory. As the author points out, this chapter has been included for the convenience of the less experienced reader. Chapter 5 is devoted to the classical applications of Galois theory. The material presented here is concerned with general constructibility criteria, cyclotomic and cyclic field extensions, algebraic equations of degree less than five, solving equations by radicals in general, computational aspects of Galois groups, and Hilbert’s irreducibility theorem in its different versions. In the final chapter, Chapter 5, the author gives an introduction to some of the fundamentals of differential Galois theory. This is certainly another particular feature of the very special algebra text under review, as here the analytic aspects of the general idea of the Galois correspondence are conveyed in very instructive a manner, and in addition to the purely algebraic point of view. The reader meets here differential fields, differential extensions, and a construction method for derivations, before getting acquainted with differential equations over a differential field in the sequel. This leads to differential Galois groups and to the differential Galois correspondence, in their general aspects, and to elementary Picard-Vessiot theory in particular. The discussion culminates in a proof of Liouville’s theorem on integration in finite terms. The algebraic setting of this analytic result in terms of differential fields is due to Ostrowski (1946), and an algebraic proof of it constitutes the concluding topic of the present book. Also, at the end of this chapter, there is an appendix providing proofs of the different versions of Hilbert’s Nullstellensatz, as this crucial tool is used several times in the course of the main text.
Each chapter comes with a large number of exercises complementing the respective material. Varying in their degree of abstraction and difficulty, many of these exercises lead the reader to related additional concepts and to further important theorems, apart from the streamline of the text. Although being quite challenging for the novice, these theoretical exercises are organized and composed in a way that should make them manageable for a smart and eager reader. Together with the exercises, this field guide to algebra is actually a much more comprehensive textbook than its title might suggest, and a great incentive to creative learning likewise. Altogether, the entire exposition stands out by its high degree of originality. Written in an utmost lucid and inspiring style, this comparatively small textbook offers a wealth of fundamental material, including all necessary prerequisites from group theory, ring theory, and differential algebra, thereby making the main text largely self-contained. The liveliness of the book is enhanced by numerous pictures from the history of mathematics, including scans of beautiful mathematical stamps and portraits of famous mathematicians. No doubt, this fairly unique introduction to some central aspects of modern abstract algebra and its applications is a highly welcome and valuable complement to the existing textbook literature in the field.