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$({\mathcal{G}}_E)\le \mathrm{ddim}(E)$. It proves the existence of quasifinite fields ${\Phi }_q:q\in \mathbb{P}$, 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 $k=0\ne \ell$.

19 pages, LaTeX: Incorporates referee's suggestions; references updated; to appear in Analele Sci. ale Univ. Ovidius, Constanta, Seria Matematica