By N. Bourbaki

Ce deuxiÃ¨me quantity du Livre d AlgÃ¨bre, deuxiÃ¨me Livre des Ã‰lÃ©ments de mathÃ©matique, traite notamment des extensions de corps et de l. a. thÃ©orie de Galois. Il comprend les chapitres: four. PolynÃ´mes et fractions rationnelles; five. Corps commutatifs; 6. Groupes et corps ordonnÃ©s; 7. Modules sur les anneaux principaux.

Il contient Ã©galement des notes historiques.

Ce quantity est une nouvelle Ã©dition 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).