Absolutely lq-Finite Extensions

Abstract

We describe the lower quasi-finite extensions $K/k$ of characteristic $p>0$, which are defined as follows: for every $n\in \mathbb N, k^{p^{-n}} \cap K/k$ is finite. We are especially interested in examining the absolute case. In this regard, we give necessary and sufficient condition for an absolutely lq-finite extension to be of finite size. Moreover, we show that any extension that is at the same time modular and lq-finite is of finite size. Furthermore, we construct an example of extension $K/k$ of infinite size such that for any intermediate field L of $K/k, L$ is of finite size over k.