Algèbre

This is a softcover reprint of the English translation of 1990 of the revised and expanded version of Bourbaki's, Algèbre, Chapters 4 to 7 (1981).
This completes Algebra, 1 to 3, by establishing the theories of commutative fields and modules over a principal ideal domain. Chapter 4 deals with polynomials, rational fractions and power series. A section on symmetric tensors and polynomial mappings between modules, and a final one on symmetric functions, have been added. Chapter 5 was entirely rewritten. After the basic theory of extensions (prime fields, algebraic, algebraically closed, radical extension), separable algebraic extensions are investigated, giving way to a section on Galois theory. Galois theory is in turn applied to finite fields and abelian extensions. The chapter then proceeds to the study of general non-algebraic extensions which cannot usually be found in textbooks: p-bases, transcendental extensions, separability criterions, regular extensions. Chapter 6 treats ordered groups and fields and based on it is Chapter 7: modules over a p. i. d. studies of torsion modules, free modules, finite type modules, with applications to abelian groups and endomorphisms of vector spaces. Sections on semi-simple endomorphisms and Jordan decomposition have been added.
Chapter IV: Polynomials and Rational Fractions
Chapter V: Commutative Fields
Chapter VI: Ordered Groups and Fields
Chapter VII: Modules Over Principal Ideal Domains.

Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements.

Ce deuxième volume du Livre d'Algèbre, deuxième Livre des Éléments de mathématique, traite notamment des extensions de corps et de la théorie de Galois. Il comprend les chapitres: 4. Polynômes et fractions rationnelles; 5. Corps commutatifs; 6. Groupes et corps ordonnés; 7. Modules sur les anneaux principaux.

Il contient également des notes historiques.

Ce volume est une nouvelle édition parue en 1981.

Polynômes et fractions rationnelles. - Corps commutatifs. - Groupes et corps ordonnés. - Modules sur les anneaux principaux.