Using Galois ideals for computing relative resolvents
Philippe Aubry, Annick Valibouze · 1998
. In this paper we establish that some ideals which occur in Galois theory are generated by a triangular set of polynomials. This geometric property seems important for the development of symbolic methods in Galois theory. It may be exploited in order to obtain more efficient algorithms. Actually, it enables us to present here a new algebraic method for computing relative resolvents which works with any kind of invariant. 1. Introduction Let k be a perfect field and ¯ k an algebraic closure of k. Let f be a univariate polynomial of k[X] supposed separable with degree n, and\\Omega be an ordered set of the n roots of f in ¯ k n . In [25] is introduced the notion of ideal of\\Omega\\Gamma32615/-39 invariant by a subset L of the symmetric group of degree n. It generalizes the notion of ideal of relations and the notion of ideal of symmetric relations. We call them Galois ideals. This paper presents two important results. First, we prove in Theorem 5.5 that a Galois ideal associated w...