The Klein resolvent

Abstract

Let K/k be a finite Galois extension of an infinite field k with group G. Let χ be an faithful (m + 1)-dimensional projective representation of G, and let Br(K/k) be the Brauer group of the extension K/k. It is proved that there exist points ξ=ξ=(ξ₁;...;ξ_m;ξ_m+1=1) in P^m(K) such that ξ^σ=ξ^ϰ(σ), ∀σ, ∈ G and K=k(ξ₁,...,ξ_m, if and only if the associated class of cohomologies η_ϰ vanishes under the homomorphism H²(G,K^*)_m+1 → Br(K/k)_m+1. We denote by ξ^σ the coordinatewise action of G, and by ξ^ϰ(σ) the geometric action determined by the representation ϰ. A construction is given for the elements ξ. As a corollary the author obtains a description of the solutions of a large class of inverse problems in Galois theory with certain constraints on ϰ and K/k.