Quasifinite fields of prescribed characteristic and Diophantine dimension

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, 𝒢_E the absolute Galois group of E, and cd(𝒢_E) the Galois cohomological dimension 𝒢_E. The research presented in this paper is motivated by the open problem of whether cd(𝒢_E) ≤ ddim(E). It proves the existence of quasifinite fields Φ_q : q ∈ ℙ, with ddim(Φ_q) infinity and char(Φ_q) = q, for each q. It shows that for any integer m > 0 and q ∈ ℙ, there is a quasifinite field Φ_m,q such that char(Φ_m,q) = q and ddim(Φ_m,q) = m. This is used for proving that for any q ∈ ℙ and each pair k, ℓ ∈ (𝕅 ∪ {0, ∞}) satisfying k ≤ ℓ, there exists a field E_k,ℓ;q with char(E_k,ℓ;q) = q, ddim(E_k,ℓ;q) = ℓ and cd(𝒢_Ek,ℓ;q) = k. Finally, we show that the field E_k,ℓ;q can be chosen to be perfect unless k = 0 ≠ = ℓ.