A quantitative version of a lemma of shafarevich-ishkhanov
Roy Meshulam, Jack Sonn · Communications in Algebra · 1999
Let q be a fixed prime power, let F = F q denote the field with q elements, and let Vn =Fn denote the space of n-tuples over F. For integers r, s,m ≥ 1 let Nq (r,s,m) denote the minimal n such that for any there exists a surjective linear map ϖ : Vn → Vr such that ϖ⊗s (wl)for all 1≥l≥ m The finiteness of Nq (r, s, m) is a key lemma used in Ishkhanov’s proof [1] of Shafarevich’s theorem on the readability of finite solvable groups as Galois groups over number fields. Here we give a new and simple proof of this lemma, as well as the following quantitative version of it: .