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Exploratory Galois theory / John Swallow

Auteur principal : Swallow, John, 1970-, AuteurType de document : MonographieLangue : anglais.Pays : Grande Bretagne.Éditeur : Cambridge : Cambridge University Press, 2004Description : 1 vol. (XII-208 p.) ; 25 cmISBN : 0521836506.Bibliographie : Bibliogr. p. 201-204. Index.Sujet MSC : 12F10, Field theory and polynomials - Field extensions, Separable extensions, Galois theory
12Fxx, Field theory and polynomials - Field extensions
12-01, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory
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The book under review provides a very first introduction to the ideas and methods of Galois theory. As the title suggests, its pedagogical goal is very special, and its methodological strategy also differs from those underlying other introductory texts on the subject. More precisely, the author's intention is to develop elementary Galois theory in as accessible a manner as possible for undergraduates, and that by an exploration-based approach. In other words, the text aims at building intuition and insight by experimenting with concrete examples from the theory of algebraic numbers, thereby presenting a Galois theory balanced between abstract concepts and computational methods. Methodologically, the text grounds the presentation in the concept of algebraic numbers with complex approximations, assuming just a modest background knowledge of abstract algebra, and the author develops the theory around elementary questions about algebraic numbers. This is done in a slow and leisurely, however direct and perspective-oriented way, together with an introduction to those technological tools for hands-on experimentation that help students to acquire a more profound familiarity with concrete number fields. Although working exclusively over the field of complex numbers, and number fields therein, the author outlines, at the end of the book, the generalization of Galois theory to arbitrary fields for the purpose of further study. The theoretical part of the exposition culminates in a discussion of the Galois correspondence for subfields of ℂ and its classical applications such as cyclotomic extensions, solvability questions for polynomial equations, and ruler-and-compass constructions. The practical part ends with explicit computations of concrete Galois groups and Galois resolvents using “Maple” or “Mathematica” as technological tools. As to the contents, the book consists of six chapters, each of which is subdivided into five or more sections. Chapter 1 provides the necessary preliminaries from abstract algebra: polynomials, polynomial rings, prime factorization in polynomial rings, computational methods, general rings and fields, general groups and, in particular, symmetric groups. Chapter 2 discusses algebraic numbers and subfields generated by algebraic numbers, together with their computational aspects. Chapter 3 illustrates the practical work with algebraic numbers and number fields generated by one algebraic number, whereas Chapter 4 deals with multiply generated number fields. Again, the computational (or exploratory) aspects, in particular the working with “Maple” or “Mathematica”, are strongly emphasized. Chapter 5 turns to normal field extensions, splitting fields, Galois groups invariant polynomials, resolvents, and discriminants, together with the related computational methods and many concrete examples. Chapter 6 is devoted to some of the celebrated classical applications as mentioned above. This chapter concludes with an outlook to Galois theory over arbitrary fields of any characteristic. Historical notes and an appendix on subgroups of symmetric groups (as used in the text) conclude the entire exposition. The rich bibliography is ordered by topics and historical relevance, which is just as helpful as the vast amount of examples and well-guided exercises throughout the whole book. Altogether, this is an excellent introduction to elementary Galois theory for very beginners. The exploration-based approach to the subject is very down-to-earth, entertaining, motivating, encouraging and enlightening, making the text highly suitable for undergraduate courses, for seminars, or for self-paced independent study by interested beginners. The computer-oriented practical aspects are rather novel, in this context and for a textbook of this kind, and they are just as timely as enhancening with regard to the current textbook literature. (Zentralblatt)

Bibliogr. p. 201-204. Index

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