Quasifinite fields of prescribed characteristic and Diophantine dimension

Abstract

Let P be the set of prime numbers, P the union P ∪ {0}, and for any field E, let char(E) be its characteristic, ddim(E) the Diophantine dimension of E, GE the absolute Galois group of E, and cd(GE) the Galois cohomological dimension of GE. This paper proves the existence of quasifinite fields Φq : q ∈ P, with ddim(Φq) infinity and char(Φq) = q, for each q. Also, it shows that for any integer m > 0 and q ∈ P, there is a quasifinite field Φm,q such that char(Φm,q) = q and ddim(Φm,q) = m. This is used for proving that for any q ∈ P and each pair k ∈ N, ℓ ∈ (N ∪ {∞}) satisfying k ≤ ℓ, there exists a perfect field E k,ℓ;q with char(E k,ℓ;q ) = q, ddim(E k,ℓ;q ) = ℓ and cd(GE k,ℓ;q ) = k.