Algebra II

Chapters 4–7

verfasst von: Nicolas Bourbaki

Verlag: Springer Berlin Heidelberg

Buchreihe : Elements of Mathematics

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten

Über dieses Buch

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

Inhaltsverzeichnis

Frontmatter

Chapter IV. Polynomials and rational fractions

Abstract

Let I be a set. We recall (III, p. 452) that the free commutative algebra on I over A is denoted by A[(X_i)_{i ∈ I}] or A[X_i]_{i ∈ I}. The elements of this algebra are called polynomials with respect to the indeterminates X_i (or in the indeterminates X_i) with coefficients in A. Let us recall that the indeterminate X_i is the canonical image of i in the free commutative algebra on I over A; sometimes it is convenient to denote this image by another symbol such as X_i̍, Y_i, T_i, etc. This convention is often introduced by a phrase such as: « Let Y = (Y_i)_{i ∈ I} be a family of indeterminates »; in this case the algebra of polynomials in question is denoted by A[Y]. When I = {1,2,..., n}, one writes A[X₁, X₂,..., X_n] in place of A[(X_i)_{i ∈ I}].

Nicolas Bourbaki

Chapter V. Commutative Fields

Abstract

Except where the contrary is expressly stated, all the fields considered in this chapter are commutative; all algebras are associative and unital and the algebra homomorphisms are unital, every subalgebra of an algebra contains the unit element of that algebra. Whenever a field K is said to be contained in a ring L (in particular in a field) without further specification, it is understood that K is a subring of L; we shall also say that K is a sub field of L, or also (if L is a field) that L is an extension field of K.

Nicolas Bourbaki

Chapter VI. Ordered groups and fields

Abstract

The notions and results presented in this section concern the study of order relations in commutative monoids (I, p. 12, Def. 2), the most important case being that of abelian groups. Unless explicitly stated otherwise, we will use additive notation for the composition law in all groups and monoids under study. On the other hand, as we go along we will present certain important algebraic applications of the theory of ordered groups and monoids, and we will accordingly translate certain of our results into the multiplicative notation which is appropriate for these applications.

Nicolas Bourbaki

Chapter VII. Modules over principal ideal domains

Abstract

Recall (I, p. 104) that an ideal of a commutative ring A is said to be principal if it has the form (a)= Aa for some a ∈ A.

Nicolas Bourbaki

Backmatter

Titel: Algebra II
verfasst von: Nicolas Bourbaki
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-61698-3
Print ISBN: 978-3-540-00706-7
DOI: https://doi.org/10.1007/978-3-642-61698-3

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter IV. Polynomials and rational fractions

Chapter V. Commutative Fields

Chapter VI. Ordered groups and fields

Chapter VII. Modules over principal ideal domains

Backmatter