Algebra II: Chapters 4–7

457 Pages·2003·9.89 MB·English

by  Nicolas Bourbaki (auth.)

ELEMENTS OF MATHEMATICS
Springer
Berlin
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NICOLAS BOURBAKI
Algebra II
Chapters 4-7
Translated by P.M. Cohn & J. Howie
, Springer
Originally published as
ALGEBRE, CHAPITRES 4 A 7
Masson, Paris, 1981
Mathematics Subject Classification (2000):
06-XX, 1OXX, 20XX, 1SXX, 18XX
Cataloging-in-Publication Data applied for
A catalog record for this book is available from the Library of Congress.
Bibliographic information published by Die Deutsche Bibliothek
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detailed bibliographic data is available in the Internet at http://dnb.ddb.de
ISBN-13: 978-3-540-00706-7 e-ISBN-13: 978-3-642-61698-3
DOl: 10.1 007/978-3-642-61698-3
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To the reader
1. The Elements of Mathematics Series takes up mathematics at the beginning,
and gives complete proofs. In principle, it requires no particular knowledge of
mathematics on the readers' part, but only a certain familiarity with mathematical
reasoning and a certain capacity for abstract thought. Nevertheless, it is directed
especially to those who have a good knowledge of at least the content of the first
year or two of a university mathematics course.
2. The method of exposition we have chosen is axiomatic, and normally' proceeds
from the general to the particular. The demands of proof impose a rigorously fixed
order on the subject matter. It follows that the utility of certain considerations will
not be immediately apparent to the reader unless he has already a fairly extended
knowledge of mathematics.
3. The series is divided into Books and each Book into chapters. The Books
already published, either in whole or in part, in the French edition, are listed below.
When an English translation is available, the corresponding English title is men
tioned between parentheses. Throughout the volume a reference indicates the English
edition, when available, and the French edition otherwise.
Theorie des Ensembles (Theory of Sets) designated by E (Set Theory)
Algebre (Algebra(1» A (Alg)
Topologie Generale (General Topology) TG (Gen. Top.)
Fonctions d'une Variable Reelle FVR
Espaces Vectoriels Topologiques (Topological
Vector Sp!lces) EVT (Top. Vect. Sp.)
Integration INT
Algebre Commutative (Commutative Algebra(2» AC (Comm. Alg.)
Varietes Differentielles et Analytiq ues VAR
Groupes et Algebres de Lie (Lie Groups and
Lie Algebras(3» LIE (Lie)
Theories Spectrales TS
e)
So far, chapters I to VII only have been translated.
e)
So far, chapters I to VII only have been translated.
e)
So far, chapters I to III only have been translated.
VI ALGEBRA
In the first six books (according to the above order), every statement in the text
assumes as known only those results which have already been discussed in the same
chapter, or in the previous chapters ordered as follows: E ; A, chapters I to III ;
TG, chapters I to III ; A, from chapters IV on ; TG, from chapter IV on ; FVR ;
EVT; INT.
From the seventh Book on, the reader will usually find a precise indication of its
logical relationship to the other Books (the first six Books being always assumed
to be known).
4. However we have sometimes inserted examples in the text which refer to facts
the reader may already know but which have not yet been discussed in the series.
Such examples are placed between two asterisks: *. .. *. Most readers will undoub
tedly find that these examples will help them to understand the text. In other cases,
the passages between *. .. * refer to results which are discussed elsewhere in the
Series. We hope the reader will be able to verify the absence of any vicious circle.
5. The logical framework of each chapter consists of the definitions, the axioms,
and the theorems of the chapter. These are the parts that have mainly to be borne
in mind for subsequent use. Less important results and those which can easily be
deduced from the theorems are labelled as « propositions », « lemmas », « corolla
ries », « remarks », etc. Those which may be omitted at a first reading are printed in
small type. A commentary on a particularly important theorem appears occasionally
under the name of « scholium ».
To avoid tedious repetitions it is sometimes convenient to introduce notations or
abbreviations which are in force only within a certain chapter or a certain section
of a chapter (for example, in a chapter which is concerned only with commutative
rings, the word« ring» would always signify «commutative ring »). Such conventions
are always explicitly mentioned, generally at the beginning of the chapter in which
they occur.
6. Some passages in the text are designed to forewarn the readZer a gainst serious
errors. These passages are signposted in the margin with the sign (<< dangerous
bend »).
7. The Exercises are designed both to enable the reader to satisfy himself that he
has digested the text and to bring to his notice results which have no place in the
text but which are nonetheless of interest. The most difficult exercises bear the sign If.
8. In general, we have adhered to the commonly accepted terminology, except
where there appeared to be good reasons for deviating from it.
9. We have made a particular effort always to use rigorously correct language,
without sacrificing simplicity. As far as possible we have drawn attention in the
text to abuses of language, without which any mathematical text runs the risk of
pedantry, not to say unreadability.
10. Since in principle the text consists of the dogmatic exposition of a theory,
it contains in general no references to the literature. Bibliographical references are
TO THE READER VII
gathered together in Historical Notes. The bibliography which follows each historical
note contains in general only those books and original memoirs which have been
of the greatest importance in the evolution of the theory under discussion. It makes
no sort of pretence to completeness.
As to the exercises, we have not thought it worthwhile in general to indicate
their origins, since they have been taken from many different sources (original
papers, textbooks, collections of exercises).
11. In the present Book, references to theorems, axioms, definitions, ... are given
by quoting successively:
- the Book (using the abbreviation listed in Section 3), chapter and page, where
they can be found;
- the chapter and page only when referring to the present Book.
The Summaries of Results are quoted by the letter R; thus Set Theory, R signifies
« Summary of Results of the Theory of Sets ».
CHAPTER IV
Polynomials and rational fractions
Throughout this chapter A denotes a commutative ring.
§ 1. POLYNOMIALS
1. Definition of polynomials
Let I be a set. We recall (III, p. 452) that the free commutative algebra on I
over A is denoted by A [(Xi)i I] or A [Xi 1 I. The elements of this algebra are
E E
called polynomials with respect to the indeterminates Xi (or in the indeterminates
XJ with coefficients in A. Let us recall that the indeterminate Xi is the canonical
image of i in the free commutative algebra on lover A; sometimes it is
convenient to denote this image by another symbol such as Xi, Yi, Ti, etc. This
convention is often introduced by a phrase such as : « Let Y = (Yi)i E I be a family
of indeterminates » ; in this case the algebra of polynomials in question is denoted
by A[Y]. When 1= {1,2, ... ,n}, one writes A[X1,XZ' ••• ,Xn] in place of
A[(Xi)i d·
E
For v E N(I) we put
TI
XV = X~i.
i E I
Then (XV)v is a basis of the A-module A [(Xi )i d. The Xv are called
E N(I) ° E
monomials in the indeterminates Xi. For v = we obtain the unit element of
A [(Xi)i I]· Every polynomial u E A [(Xi)i I] can be written in exactly one way
E E
in the form
where CXv E A and the CXv are zero except for a finite number ; the CXv are called the
coefficients of u; the cxvXv are called the terms of u (often the element
cxvXv is called the term in XV), in particular the term of cxoXo, identified with
CXo, is called the constant term of u. When CXv = 0, we say by abuse of language that
u contains no element in Xv; in particular when CXo = 0, we say that u is a
polynomial without constant term (III, p. 453). Any scalar multiple of 1 is called a
constant polynomial.
A.IV.2 POLYNOMIALS AND RATIONAL FRACTIONS § 1
Let B be a commutative ring and p : A --> B a ring homomorphism. We consider
B [(Xi )i as an A-algebra by means of p. Thus the mapping a of A [(Xi )i into
E [] E []
L L
B [(Xi )i which transforms ex.X· into p (ex. )X· is a homomorphism of A-
E []
algebras ; if U E A [ (Xi )i E [ ], we sometimes denote by Pu the image of u by this
homomorphism. The homomorphism of B ® A A [(Xi )i E [] into B [(Xi )i E [ ]
canonically defined by a transforms, for every i E I, 1 ® Xi into Xi ; this is an
isomorphism of B-algebras (III, p. 449).
Let M be a free A-module with basis (ei)i There exists precisely one unital
E [.
homomorphism 'P of the symmetric algebra S (M) into the algebra A [(Xi )i E [ ]
such that 'P (ei) = Xi for each i E I, and this homomorphism is an isomorphism
(III, p. 506). This isomorphism is said to be canonical. It allows us to apply to
polynomial algebras certain properties of symmetric algebras. For example, let
[.1
(I>JA E L be a partition of I. Let 'PA be the homomorphism of P A = A [(Xi )i E into
P = A [(Xi)i E r1 which transforms Xi (qua element of PA) into Xi (qua element of
P). Then the 'PA define a homomorphism of the algebra ® PA into the algebra P,
A E L
and this homomorphism is an isomorphism (III, p. 503, Prop. 9).
Let E be an A-module, and put E ® A A [(Xi)i E [] = E [(Xi)i E []. The elements
of the A-module E [(Xi )i are called polynomials in the indeterminates
E []
Xi with coefficients in E. Such a polynomial can be written in just one way as
L
e. ® X·, where e. E E and the e. are zero for all but a finite number of
suffixes; we frequently write e.X· instead of e. ® Xv.
2. Degrees
Let P = A [(Xi )i E [] be a polynomial algebra. For each integer n E N let
P n be the submodule of P generated by the monomials X· such that Iv I = L Vi
i E I
equals n. Then (P n)n E N is a graduation which turns A [(Xi)i E [] into a graded
algebra of type N (III, p. 459). The homogeneous elements of degree n in
A [(Xi)i are sometimes called forms of degree n with respect to the indetermi
E []
nates Xi.
When we are dealing with the degree of inhomogeneous polynomials, we shall
agree to adjoin to the set N of natural numbers, an element written - 00 and to
extend the order relation and the addition of N to N U {- oo} by the following
conventions, where n E N,
-oo<.n, (-oo)+n=n+(-oo)=-oo, (-00)+(-00)=-00.
L
Let u = ex.X· be a polynomial. The homogeneous component Un of degree
ve N(ll