Galois extensions, positive involutions and an application to unitary space-time coding

TL;DR

This paper investigates the structure of maximal symmetric subfields in central division algebras with positive involutions, establishing conditions for Galois extensions and applying these results to optimize constructions in unitary space-time coding.

Contribution

It proves that certain maximal symmetric subfields are Galois extensions and shows that conditions for positive involutions in division algebras are both necessary and sufficient, confirming the optimality of Berhuy's construction.

Findings

Maximal symmetric subfields are Galois extensions under certain conditions.

Necessary and sufficient conditions for positive involutions in specific division algebras.

Berhuy's construction in unitary space-time coding is proven to be optimal.

Abstract

We show that under certain conditions every maximal symmetric subfield of a central division algebra with positive unitary involution $(B,\tau)$ will be a Galois extension of the fixed field of $\tau$ and will "real split" $(B,\tau)$. As an application we show that a sufficient condition for the existence of positive involutions on certain crossed product division algebras, considered by Berhuy in the context of unitary space-time coding, is also necessary, proving that Berhuy's construction is optimal.