Symbol algebras and cyclicity of algebras after a~scalar extension

U. Rehmann ; S. V. Tikhonov ; V. I. Yanchevskii

Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 6, pp. 193-209

Résumé

For a field $F$ and a family of central simple $F$-algebras we prove that there exists a regular field extension $E/F$ preserving indices of $F$-algebras such that all the algebras from the family are cyclic after scalar extension by $E$. Let $\mathcal A$ be a central simple algebra over a field $F$ of degree $n$ with a primitive $n$th root of unity $\rho_n$. We construct a quasi-affine $F$-variety $\mathrm{Symb}(\mathcal A)$ such that for a field extension $L/F$ $\mathrm{Symb}(\mathcal A)$ has an $L$-rational point if and only if $\mathcal A\otimes_FL$ is a symbol algebra. Let $\mathcal A$ be a central simple algebra over a field $F$ of degree $n$ and $K/F$ be a cyclic field extension of degree $n$. We construct a quasi-affine $F$-variety $C(\mathcal A,K)$ such that, for a field extension $L/F$ with the property $[KL:L]=[K:F]$, the variety $C(\mathcal A,K)$ has an $L$-rational point if and only if $KL$ is a subfield of $\mathcal A\otimes_FL$.