# The cyclicity problem for Albert algebras

## TL;DR

This paper proves that every Albert division algebra over any field contains a cyclic cubic subfield in some isotope, and its structure group includes a subgroup of type $^3D_4$, advancing understanding of their internal structure.

## Contribution

It demonstrates that for any Albert division algebra, an isotope exists with a cubic cyclic subfield, and the structure group contains a specific subgroup, addressing the cyclicity problem.

## Key findings

- Existence of a cyclic cubic subfield in some isotope of any Albert division algebra
- Structure group contains a subgroup of type $^3D_4$ over the base field
- Advances the solution to the Albert cyclicity problem

## Abstract

In this paper we address the celebrated Albert problem for exceptional Jordan algebras (i.e. Albert algebras): Does every Albert division algebra contain a cubic cyclic subfield? We prove that for any Albert division algebra $A$ over a field $k$ of arbitrary characteristic, there is a suitable isotope that contains a cubic cyclic subfield. It follows from this that for any Albert division algebra $A$ over a field $k$, the structure group $\text{\bf Str}(A)$ always contains a subgroup of type $^3D_4$ defined over $k$.