Quasifinite fields of prescribed characteristic and Diophantine dimension

Let $\mathbb{P}$ be the set of prime numbers, $\bar{\mathbb{P}}$ the union $\mathbb{P}\cup \{0\}$, and for any field $E$, let char$(E)$ be its characteristic, ddim$(E)$ the Diophantine dimension of $E$, ${\mathcal{G}}_E$ the absolute Galois group of $E$, and cd$({\mathcal{G}}_E)$ the Galois cohomological dimension ${\mathcal{G}}_E$. The research presented in this paper is motivated by the open problem of whether cd$({\mathcal{G}}_E)\le \mathrm{d}\mathrm{d}\mathrm{i}\mathrm{m}(E)$. It proves the existence of quasifinite fields ${\mathrm{\Phi }}_q\text{\hspace{0.17em}⁣}:q\in \mathbb{P}$, with ddim$({\mathrm{\Phi }}_q)$ infinity and char$({\mathrm{\Phi }}_q)=q$, for each $q$. It shows that for any integer $m>0$ and $q\in \bar{\mathbb{P}}$, there is a quasifinite field ${\mathrm{\Phi }}_{m,q}$ such that char$({\mathrm{\Phi }}_{m,q})=q$ and ddim$({\mathrm{\Phi }}_{m,q})=m$. This is used for proving that for any $q\in \bar{\mathbb{P}}$ and each pair $k$, $\mathrm{\ell }\in (\mathbb{N}\cup \{0,\mathrm{\infty }\})$ satisfying $k\le \mathrm{\ell }$, there exists a field $E_{k,\mathrm{\ell };q}$ with char$(E_{k,\mathrm{\ell };q})=q$, ddim$(E_{k,\mathrm{\ell };q})=\mathrm{\ell }$ and cd$({\mathcal{G}}_{E_{k,\mathrm{\ell };q}})=k$. Finally, we show that the field $E_{k,\mathrm{\ell };q}$ can be chosen to be perfect unless $k=0\mathrm{\ne }\mathrm{\ell }$.