$p$th powers of distinguished subfields

Abstract:

Let $k \supset s \supset {k_0}$ be fields of characteristic $p \ne 0,k/{k_0}$ finitely generated and $s$ a distinguished subfield. The field ${k_0}({k^{(n)}}) = \{ x \in k|{x^{{p^t}}} \in {k_0}({k^{{p^{n + t}}}}){\text { for some }}t \geqslant 0\}$ has ${k_0}({s^{{p^n}}})$ as a distinguished subfield and is maximal in $k$ with respect to this property. Let $\overline k _0^s$ and ${\bar k_0}$ be, respectively, the separably algebraic closure and the algebraic closure of ${k_0}$ in $k$. Then ${\overline k _0} = { \cap _k}{k_0}({k^{(n)}})$. Also $\overline k _0^s = {\bar k_0}$ if and only if ${k_0}({k^p}) \supset {k_0}({k^{(n)}})$ for some $n$. For $n$ large ${k_0}({k^{(n)}}) = {\overline k _0}({k^{{p^n}}})$. The sequence ${\{ {[{k_0}({k^{(n)}}):{k_0}]_i}\} _n}$ is decreasing, descending from ${[k:{k_0}]_i}\;{\text {to }}{[\overline {k:} {k_0}]_i}$ in a finite number of steps. Examples are given which show: (1) that ${k_0}({k^{(n)}})$ may have distinguished subfields not of the form ${k_0}({s^{{p^n}}})$; and, (2) how to construct $k/{k_0}$ so that the sequence $\{ [{k_0}{({k^{(n)}}:{k_0}]_i}\}$ has preassigned values.