Totally ramified subfields of $p$-algebras over discrete valued fields with imperfect residue

TL;DR

This paper proves a conjecture regarding the existence of totally ramified cyclic maximal subfields in $p$-algebras over complete discrete valued fields with imperfect residue fields, under certain conditions.

Contribution

It establishes the conjecture that such $p$-algebras contain a totally ramified cyclic maximal subfield if they contain a totally ramified purely inseparable maximal subfield, given specific conditions on the residue field.

Findings

Proves the conjecture for imperfect residue fields.

Shows the existence of cyclic maximal subfields under certain conditions.

Extends understanding of $p$-algebras over valued fields.

Abstract

Let $K$ be a complete discrete valued field of characteristic $p$ with residue $k$ which is not necessarily perfect. We prove the Conjecture in \cite{cs} that a $p$-algebra over $K$ contains a totally ramified cyclic maximal subfield if it contains a totally ramified purely inseparable maximal subfield provided $k$ satisfies some conditions on its $p$-rank.