Quasifinite fields of prescribed characteristic and Diophantine dimension

Abstract

Let $\mathbb{P}$ be the set of prime numbers, $\overline {\mathbb{P}}$ 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$(\mathcal{G}_{E}) \le {\rm ddim}(E)$. It proves the existence of quasifinite fields $\Phi _{q}\colon q \in \mathbb{P}$, with ddim$(\Phi _{q})$ infinity and char$(\Phi _{q}) = q$, for each $q$. It shows that for any integer $m > 0$ and $q \in \overline {\mathbb{P}}$, there is a quasifinite field $\Phi _{m,q}$ such that char$(\Phi _{m,q}) = q$ and ddim$(\Phi _{m,q}) = m$. This is used for proving that for any $q \in \overline {\mathbb{P}}$ and each pair $k$, $\ell \in (\mathbb{N} \cup \{0, \infty \})$ satisfying $k \le \ell $, there exists a field $E _{k, \ell ; q}$ with char$(E _{k, \ell ; q}) = q$, ddim$(E _{k, \ell ; q}) = \ell $ and cd$(\mathcal{G}_{E_{k, \ell ; q}}) = k$. Finally, we show that the field $E _{k, \ell ; q}$ can be chosen to be perfect unless $k = 0 \neq \ell $.