Using Galois Ideals for Computing Relative Resolvents

Let k be a perfect field and k̄ an algebraic closure of k. Let f be a separable univariate polynomial of k[X] with degree n, and Ω be an ordered set of the n distinct roots of f in k̄. In Valibouze (1999) the notion of ideal of Ω-relations invariant by a subset L of the symmetric group of degree n is introduced. 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 major results. First, we prove in Theorem 5.3 that a Galois ideal associated with a group L containing the Galois group of f is generated by a separable triangular set of polynomials which forms a reduced Gröbner basis of this ideal for the lexicographical order. We think knowledge of such a property may simplify some problems, and thus it may be a basic tool for making algorithms in Galois theory more efficient. This remark may be taken into account when one is concerned with optimal implementation issues. Moreover, it may lead to new algorithms in Galois theory. The second major result of this paper illustrates this assertion since an algebraic method for computing relative resolvents is given (see Section 7). We also specify (Remark 7.11) that when the Galois group of the polynomial f is known, our algorithms may be employed for computing the ideal of relations among the roots of f and consequently for computations in the splitting field of f . The resolvent is a fundamental tool, introduced by Lagrange (1770), in the constructive Galois theory. Later, Stauduhar (1973) extended the definition of J. L. Lagrange. Let us recall that the resolvents relative to the symmetric group Σn, called absolute resolvents, can be computed by many algorithms (Lagrange, 1770; Soicher, 1981; Soicher and McKay, 1985; Valibouze, 1989a; Casperson and McKay, 1994). For computing resolvents relative to some proper subgroup L of Σn, there exists a numerical approach (Stauduhar, 1973; Eichenlaub, 1996) and a p-adic approach (Darmon and Ford, 1989; Yokoyama, 1996; Geissler and Klüners, 2000) used for Galois group computation based on Stauduhar’s algorithm. But in these methods, we do not need the resolvents themselves; we compute their integral roots with approximations of the roots of the given