N. Bourbaki's Algèbre: Chapitre 4 à 7 PDF

By N. Bourbaki

ISBN-10: 3540343989

ISBN-13: 9783540343981

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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Additional resources for Algèbre: Chapitre 4 à 7

Sample text

40. Ainsi, (ii) est démontré. En particulier, ~ l ( ~ 2= ~ 3T )~ G ~ , + ~ , + ~O,Z,2 ' OQ~~3,r )~ ~ , ~ ~ ~ ( z ~ et l'on établit de même que Donc l'algèbre TS(M) est associative. Soit o l'élément de G, +,, tel que Alors Donc l'algèbre TS(M) est comrrutative. On prendra garde que l'injection canonique de TS(M) dans T(M) n'est pas en général un homomorphisme d'algèbres. Pis encore, TS(M) n'est pas en général stable par la multiplication de T(M). 4. Puissances divisées Soient x E M et k E N . Il est clair que x, O x, O ...

Sn-,. D'après ( B ) , il existe un endomorphisme u de la B-algèbre S' tel que D'après ( 5 ) , on a u(sk) = u(sk) + u(sk- l ) X , , d'où u(sk) = s; par un calcul facile. , Y,- ,] ; de P(s, , ... , s,- ,) = O , on déduit alors d'où P = O d'après (B). Par suite, la B-algèbre S' est engendrée par les éléments algébriquement indépendants s, , ... , sn-, . , S n - , > X,. , sn- S,, ,. Prouvons a). Soit f E S un polynôme symétrique homogène de degré m. On a ,f' E S' = C[XJ et il existe donc un élément g = P(s, , ..

On note f ( g ) ou f O g l'élément Soient f = (filiel E (A[[J]])' ( . L ( ( g j ) j s J ) ) i e t de (A[[KlI)'. Si f E AI,,, on a f 0 g E A,,, puisque l'application f Hf ( ( s ~ ) de ~ ~A[[J]] , ) dans A [ [ K ] ]est continue. Soient f E (A[[J]])',g E A,,,, h E A,,,. Alors g o h E A,,, et, d'après (4), on a 4. Séries formelles inversibles PROPOSITION 5. - Dans l'anneau A [ [ T ] ]des séries formelles en une indéterminée, 30 le polynôme 1 - T est inversible, et l'on a ( 1 - T ) - ' = C Tn. En effet, PROPOSITION 6.

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Algèbre: Chapitre 4 à 7 by N. Bourbaki

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