# Isotopy and invariants of Albert algebras

### Maneesh L. Thakur

Tata Institute of Fundamental Research, Mumbai, India

## Abstract

Let k be a field with characteristic different from 2 and 3. Let B be a central simple algebra of degree 3 over a quadratic extension K/k, which admits involutions of second kind. In this paper, we prove that if the Albert algebras $J(B,σ,u,μ)$ and $J(B,τ,v,ν)$ have same $f_{3}$ and $g_{3}$ invariants, then they are isotopic. We prove that for a given Albert algebra J, there exists an Albert algebra J' with $f_{3}(J_{′})=0$ , $f_{5}(J_{′})=0$ and $g_{3}(J_{′})=g_{3}(J)$ . We conclude with a construction of Albert division algebras, which are pure second Tits' constructions.

## Cite this article

Maneesh L. Thakur, Isotopy and invariants of Albert algebras. Comment. Math. Helv. 74 (1999), no. 2, pp. 297–305

DOI 10.1007/S000140050090