Etude des relations algébriques entre les racines d'un polynôme d'une variable
Résumé
Galois theory allows us to deal with effective computation in algebraic extensions of fields. In this aim, the present paper is devoted to an inductive construction of a generating system for the ideal of relations among the roots of a univariate polynomial over a field. The idea is to define new ideals between the ideal of symmetric relations and the ideal of relations and to give a correspondence between these ideals and finite sets of permutations. The fundamental tools of this construction are multivariate polynomials called minimal polynomials associated to our ideals. These polynomials characterize the considered ideals and allow to construct a generating system for them.
Cet article développe une vision effective de la théorie de Galois algébrique en apportant des propriétés inhérentes aux idéaux associés à un polynôme d’une variable. Il débouche sur un algorithme de calcul du groupe de Galois d’un polynôme et de l’idéal des relations entre ses racines.
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