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

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)