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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, 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
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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