Hilbert 90 for Algebras with Conjugation

TL;DR

This paper extends Hilbert 90 to a broad class of non-commutative, non-associative algebras, enabling parametrization of solutions to norm one equations in ordered fields.

Contribution

It introduces a generalized version of Hilbert 90 applicable to complex algebraic structures, including iterated Conway-Smith doublings.

Findings

Generalized Hilbert 90 for non-commutative algebras

Parametrization of norm one solutions in ordered fields

Applicable to iterated algebraic constructions

Abstract

We show a version of Hilbert 90 that is valid for a large class of algebras many of which are not commutative, distributive or associative. This class contains the nth iteration of the Conway-Smith doubling procedure. We use our version of Hilbert 90 to parametrize all solutions in ordered fields to the norm one equation for such algebras.