Common subfields of p-algebras of prime degree

TL;DR

This paper investigates the relationships between division p-algebras of prime degree and their shared inseparable and cyclic separable field extensions, revealing specific implications and limitations of these shared structures.

Contribution

It establishes that sharing an inseparable extension implies sharing a cyclic separable one, but the converse does not hold, clarifying the structure of p-algebras.

Findings

Sharing an inseparable extension implies sharing a cyclic separable extension.

The converse implication does not generally hold.

Sharing all inseparable extensions does not imply sharing all cyclic separable extensions.

Abstract

We show that if two division $p$-algebras of prime degree share an inseparable field extension of the center then they also share a cyclic separable one. We show that the converse is in general not true. We also point out that sharing all the inseparable field extensions of the center does not imply sharing all the cyclic separable ones.