Galois theory allows us to deal with effective computation in algebraic extensions of fields. In this aim, the present paper is devoted to an effective inductive construction of an effective generator system for the ideal of relations between the roots of an 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 correspondance 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 characterise the considered ideals and allow to construct a generators system for them.