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

TL;DR

The paper proves the 'if' part of the conjecture that a $p$-algebra over a complete discretely valued field $K$ with imperfect residue $k$ contains a totally ramified cyclic maximal subfield whenever it contains a totally ramified purely inseparable maximal subfield, under two conditions on $k$'s $p$-rank. The approach combines Albert's cyclic-extension construction and Witt-vector methods to lift residue extensions of exponent one, builds division algebras $[\omega,b)_K$ sharing an inseparable subfield, and uses linkage results (CFM) to extract a common cyclic maximal subfield with totally ramified behavior. The paper develops a method to produce weakly unramified cyclic extensions with prescribed residue fields and provides a concrete criterion ensuring division algebras via valuation-theoretic arguments. This advances understanding of the interplay between ramification, residue fields, and maximal subfields in $p$-algebras over imperfect residues.

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.

Theorems & Definitions (12)

• Conjecture 1.1
• Theorem 1.2
• Theorem 3.1
• Remark 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• Theorem 4.1
• proof
• Lemma 5.1