Generic Polynomials are Descent-Generic

Let g(X) ∈ K(t1, . . . , tm)[X] be a generic polynomial for a group G in the sense that every Galois extension N/L of infinite fields with group G and K ≤ L is given by a specialization of g(X). We prove that then also every Galois extension whose group is a subgroup of G is given in this way. Let K be a field and G a finite group. Let us call a monic, separable polynomial g(t1, . . . , tm, X) ∈ K(t1, . . . , tm)[X] generic for G over K if the following two properties hold. (1) The Galois group of g (as a polynomial in X over K(t1, . . . , tm)) is G. (2) If L is an infinite field containing K and N/L is a Galois field extension with group G, then there exist λ1, . . . , λm ∈ L such that N is the splitting field of g(λ1, . . . , λm, X) over L. We call g descent-generic if it satisfies (1) and the stronger property (2’) If L is an infinite field containing K and N/L is a Galois field extension with group H ≤ G, then there exist λ1, . . . , λm ∈ L such that N is the splitting field of g(λ1, . . . , λm, X) over L. DeMeyer [2] proved that the existence of an irreducible descent-generic polynomial for a group G over an infinite field K is equivalent to the existence of a generic extension S/R for G over K in the sense of Saltman [6]. Ledet [5] proved that the existence of a generic polynomial for a group G over an infinite field K is equivalent to the existence of a generic extension S/R of G over K. Thus for K infinite the existence of a generic polynomial for G implies the existence of a descent-generic polynomial for G. In this note we prove the following stronger result. Theorem 1. Every generic polynomial g(t1, . . . , tm, X) for G over K is descent-generic. Proof. G has a faithful, transitive permutation representation G ↪→ Sn, by which it acts on the rational function field K(x1, . . . , xn). K(x1, . . . , xn) is Galois over K(x1, . . . , xn) with group G, hence there exist p1, . . . , pm ∈ K(x1, . . . , xn) such that K(x1, . . . , xn) is the splitting field of f(X) := g(p1, . . . , pm, X) over K(x1, . . . , xn). Write