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

Auteur: Bourbaki, Nicolas - AuteurType de document: MonographieLangue: françaisPays: FranceÉditeur: Paris, : Masson, 1981Description: 1 vol. (VII-422 p.) ; 24 cm ISBN: 2225685746 ; rel. Note: Exercices en fin de chapitresRésumé: 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).Bibliographie: Bibliogr. p. [405-406]. Index. Sujets MSC: 12Fxx Field theory and polynomials -- Field extensions
13F10 Commutative algebra -- Arithmetic rings and other special rings -- Principal ideal rings
13C10 Commutative algebra -- Theory of modules and ideals -- Projective and free modules and ideals
12E05 Field theory and polynomials -- General field theory -- Polynomials (irreducibility, etc.)
05E05 Combinatorics -- Algebraic combinatorics -- Symmetric functions and generalizations
Location Call Number Status Date Due
Salle S 08043-01 / Séries BOU (Browse Shelf) Available

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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