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Eléments de mathématique : algèbre, Chapitres 4 à 7 / N. Bourbaki

Auteur principal : Bourbaki, Nicolas, AuteurType de document : MonographieLangue : français.Pays : France.Éditeur : Paris, : Masson, 1981Description : 1 vol. (VII-422 p.) ; 24 cmISBN : 2225685746.Bibliographie : Bibliogr. p. [405-406]. Index.Sujet MSC : 12Fxx, Field theory and polynomials - Field extensions
13F10, Arithmetic rings and other special commutative rings, Principal ideal rings
13C10, Theory of modules and ideals in commutative rings, Projective and free modules and ideals
12E05, Field theory and polynomials - General field theory, Polynomials in general fields
05E05, Algebraic combinatorics, Symmetric functions and generalizations
En-ligne : ed. Springer
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Exercices en fin de chapitres

Bibliogr. p. [405-406]. Index

Chapter 4 develops the abstract theory of general polynomial rings, function fields, and formal power series, including the differential aspects (differentials and derivations) of these topics as a fundamental part. This is enriched by an in-depth treatment of symmetric tensor algebras, divided powers, polynomial maps, and their functorial interrelations, on the one hand, and by a just as comprehensive discussion of symmetric polynomials, symmetric rational functions, symmetric power series, resultants, and discriminants, on the other. Chapter 5 is then devoted to the theory of commuative fields, their various kinds of extensions, and the allied theory of étale algebras over a ground field. Apart from the fundamentals of Galois theory, Kummer theory, Artin-Schreier theory, and of the theory of finite fields, this chapter also discusses separable algebras and differential criteria for separability in full generality and detail. Chapter 6 briefly describes the basics of ordered groups and ordered fields, together with their respective fundamental structure theorems, whereas Chapter 7 deals with the theory of modules over a principal domain and its applications to the study of endomorphisms of finite-dimensional vector spaces. (Zentralblatt)

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