Download Algèbre: Chapitre 4 à 7 by N. Bourbaki PDF

Download Algèbre: Chapitre 4 à 7 by N. Bourbaki PDF

By N. Bourbaki

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

Ce deuxième quantity du Livre d Algèbre, deuxième Livre des Éléments de mathématique, traite notamment des extensions de corps et de l. a. théorie de Galois. Il comprend les chapitres: four. Polynômes et fractions rationnelles; five. Corps commutatifs; 6. Groupes et corps ordonnés; 7. Modules sur les anneaux principaux.

Il contient également des notes historiques.

Ce quantity est une nouvelle édition parue en 1981.

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Extra info for Algèbre: Chapitre 4 à 7

Sample text

Il est clair que b, est un idéal de l'anneau A[[X]],engendré par les monômes de degré n. Par suite b, se compose des sommes finies de produits de n séries formelles sans terme constant ; si D est une dérivation de A[[X]],on a d'où immédiatement Db, c b,-, pour n 2 1. Comme la suite (b,),, est un système fondamental de voisinages de O dans A [ [ X ] ] (IV, p. 25, remarque), D est continue, d'où (i). Raisonnant comme précédemment, on montre que A(h) appartient à b,-, pour tout polynôme h homogène de degré n 2 1.

Soient M et N deux A-modules. On suppose que M est libre. Soit Ap(M, N) le A-module des applications de M dans N. Le sous-module 1PolA(M, N) qbO de Ap(M, N) se note PolA(M,N), ou simplement Pol(M, N) ; ses éléments s'appellent les applications polynomiales de M dans N. Soit (e,),,, une base de M, et supposons 1 fini; d'après la prop. 13 (IV, p. 51), une application f de M dans N est polynomiale si et seulement s'il existe un polynôme F a coefficients dans N en les indéterminées Xi tel que l'on ait pour toute famille x = (xi)i,, dans A'".

Soit O = B/H. Pour tout o E R, soit u, = b. Alors 1 b€, (i) (u,),,, est une base du A-module UH. - (ii) Pour tout o E Q , soit V , un point de o ; posons w' = o {v,} et B' = U o'. o~n Alors Br est une base d'un supplémentaire de UHdans U . La réunion de l'ensemble des u , (pour o E Q ) et de B' est une base de U. Si U' = Au, et U" = 1Ab, on a donc U = U' @ U". Enfin, soit (a,),,, une famille d'éléments de A a,b. Par suite, U' = UH. PROPOSITION 5. - Soient M un A-module libre, k un entier 3 O, P le sous-A-module de T S k ( M )engendré par yk(M).

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