(FUNDAMENTAL AND APPLIED MATHEMATICS)

2008, VOLUME 14, NUMBER 6, PAGES 193-209

## Symbol algebras and cyclicity of algebras after a scalar extension

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

Abstract

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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 rn. We construct a quasi-affine $F$-variety $Symb\left(\mathcal A\right)$ such that for a field extension $L/F$ $Symb\left(\mathcal A\right)$ has an $L$-rational point if and only if $\mathcal A \otimes_F L$ 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 $C\left(\mathcal A, K\right)$ such that, for a field extension $L/F$ with the property $\left[KL:L\right]=\left[K:F\right]$, the variety $C\left(\mathcal A, K\right)$ has an $L$-rational point if and only if $KL$ is a subfield of $\mathcal A \otimes_F L$.

Location: http://mech.math.msu.su/~fpm/eng/k08/k086/k08611h.htm