Albert's twisted field construction using division algebras with a multiplicative norm

TL;DR

This paper generalizes Albert's twisted field construction to unital division algebras with a multiplicative norm, providing conditions for the resulting algebras to remain division algebras and constructing examples from quaternions and octonions.

Contribution

It extends Albert's construction to broader classes of division algebras and identifies conditions for their division property, with explicit examples from classical algebras.

Findings

Conditions for resulting algebras to be division algebras

Construction of 4- and 8-dimensional division algebras from quaternions and octonions

Large derivation algebras in the constructed examples

Abstract

We generalize Albert's twisted field construction, applying it to unital division algebras with a multiplicative norm. We give conditions for the resulting algebras to be division algebras.Four- and eight-dimensional real unital and non-unital division algebras with large derivation algebras are constructed out of Hamilton's quaternion and Cayley's octonion algebra.