Introduction to Galois theory (Q2855398)

The book under review grew out of a lecture course that the two authors have successively given at École Polytechnique, Palaiseau, France. As they point out in the preface, the main objective of both the lectures and the present booklet is to provide first an introduction to those basic concepts and, methods of abstract algebra which are needed to develop the fundamentals of elementary Galois theory, and to treat the latter classical topic thereafter in this general framework. With regard to this special didactic conception, and in view of the fact that no advanced mathematical prerequisites are assumed, this particular introduction to Galois theory may also serve as a first introduction to modern abstract algebra and its various applications. As to the precise contents, the book consists of twelve chapters, each of which comprises several thematic subsections.NEWLINENEWLINE NEWLINEActually, the text starts with a preceding section titled ``Invitation to Galois Theory'', where a brief historical sketch of two famous applications of classical Galois theory is given: ruler-compass constructions and solutions of algebraic equations.NEWLINENEWLINE NEWLINEChapter 1 discusses some complementary facts from basic group theory, including exact sequences of group homomorphisms, group actions on sets, symmetric groups, and solvable groups.NEWLINENEWLINENEWLINEChapter 2 briefly treats some further topics in ring theory, culminating in the description of the rank of a finitely generated free module, on the one hand, and in the explanation of the Frobenius homomorphism of a ring of characteristic \(p>0\).NEWLINENEWLINENEWLINEChapter 3 develops the fundamentals on algebras over a field, thereby introducing algebraic and transcendental elements, splitting fields of polynomials, and the algebraic closure of a field.NEWLINENEWLINENEWLINEChapter 4 turns to finite fields and their automorphisms, the Berlekamp factorization algorithm for polynomials over a finite field, perfect fields and their extensions, separable extensions of perfect fields, and the theorem of the existence of primitive elements in this context.NEWLINENEWLINENEWLINEThe introduction to Galois theory starts with Chapter 5, where Galois field extensions and some of their various characterizations, Galois groups, and the Galois correspondence are explained. Cyclotomic field extensions are the topic of Chapter 6, where an application of their Galois theory to the constructibility of regular polygons (à la Gauss-Wantzel) is presented at the end.NEWLINENEWLINENEWLINEIn Chapter 7, the second historical application of Galois theory is studied, namely the criterion of solvability of polynomial equations by radicals. In the course of this chapter, the Galois group of a polynomial, discriminants, and cyclic field extensions are also discussed. The main text ends with the subsequent Chapter 8, where the method of ``reduction modulo \(p\)'' as a tool for computing the Galois groups of polynomials with integer coefficients is described in greater detail, thereby using much of the material developed in the previous, introductory chapters. No doubt, this chapter represents one of the highlights of the book and must be seen as one of its main features.NEWLINENEWLINEChapter 9 contains six appendices providing a few additional topics such as Zorn's Lemma, the transcendence of \(e\) and \(\pi\), symmetric polynomials, and inverse Galois theory.NEWLINENEWLINE NEWLINEFinally, Chapter 10 is a collection of the problems of the final exams for this lecture course covering the years from 2005 to 2011, whereas Chapter 11 gives complete solutions to these example problems.NEWLINENEWLINENEWLINEChapter 12, the last chapter of the book, provides solutions to the few exercises scattered throughout the text itself.NEWLINENEWLINE NEWLINEAll together, this utmost concise and streamlined introduction to Galois theory, its underlying abstract algebra, and its concrete applications is a perfect first reading for upper-undergraduate students in mathematics, computer science, and related areas.