A classification of the division algebras that are isotopic to a cyclic Galois field extension

TL;DR

This paper classifies division algebras that are isotopic to cyclic Galois field extensions, providing a detailed and distinguishable list especially for cubic cases, advancing understanding of algebraic structures related to Galois theory.

Contribution

It offers a comprehensive classification of principal Albert isotopes of cyclic Galois extensions, including a precise classification for cubic extensions, enhancing the understanding of their algebraic structure.

Findings

Complete classification of division algebras isotopic to cyclic Galois extensions for degrees n>2.

A 'tight' classification for cubic Galois extensions.

Explicit criteria to distinguish non-isomorphic algebras.

Abstract

We classify all division algebras that are principal Albert isotopes of a cyclic Galois field extension of degree $n>2$ up to isomorphisms. We achieve a ``tight'' classification when the cyclic Galois field extension is cubic. The classification is ``tight'' in the sense that the list of algebras has features that make it easy to distinguish non-isomorphic ones.