A hermitian analogue of the Broecker-Prestel theorem

TL;DR

This paper extends the Broecker-Prestel local-global principle to hermitian forms over certain algebras, providing new characterizations and improved results for specific algebra classes.

Contribution

It introduces a hermitian analogue of the Broecker-Prestel theorem for algebras of index at most two and enhances results for algebras with decomposable involutions and over special fields.

Findings

Hermitian analogue of the local-global principle established.

Improved results for algebras with decomposable involution.

Characterizations over SAP and ED fields achieved.

Abstract

The Broecker-Prestel local-global principle characterizes weak isotropy of quadratic forms over a formally real field in terms of weak isotropy over the henselizations and isotropy over the real closures of that field. A hermitian analogue of this principle is presented for algebras of index at most two. An improved result is also presented for algebras with a decomposable involution, algebras of pythagorean index at most two, and algebras over SAP and ED fields.