Mathematics – Logic
Scientific paper
2003-06-10
Mathematics
Logic
19 pages, in French
Scientific paper
Let k be a differential field and C its subfield of constants. In general a differential extension K of k add some new constants to C, and it is difficult to prove that C stay unchangeable under the extension K; This situation is provided by the Picard-Vessiot extension. Kolchin prove the theorem of existence and unicity for these extensions. The aim of this paper is to prove Kolchin theorem and other results, in a simple manner, by means of the theory of models and logic. ----- Soit k un corps diff\'erentiel et C son sous corps des constantes. En g\'en\'eral une extension diff\'erentiel K de k modifie le corps des constantes C de k. Prouver que K ne modifie pas C est un probl\`eme assez difficile en alg\`ebre diff\'erentiel. Les extensions de Picard-Vessiot constitue un exemple de cette situation. Kolchin a montr\'e le th\'eor\`eme d'existence et d'unicit\'e, \`a isomorphisme pr\'es, des extensions de Picard-Vessiot sous la condition que le corps C est alg\'ebriquement clos. Dans ce travail on utilise la th\'eorie des corps diff\'erentiellements clos (Th\'eorie des mod\`eles), pour montrer l'existence et l'unicit\'e, \`a isomorphisme pr\'es, des extensions de Picard-Vessiot. On calcul ensuite le groupe de Galois diff\'erentiel de certaines extensions particuli\`ere. Enfin, on montre quelque th\'eor\`emes g\'en\'eraux de la th\'eorie de Galois diff\'erentielle par les m\^emes techniques.
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