FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(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 $\backslash mathcal\; A$ be a central simple algebra over a field $F$ of degree $n$ with a primitive $n$th root of unity $$r_n. We construct a quasi-affine $F$-variety $Symb(\backslash mathcal\; A)$ such that for a field extension $L/F$ $Symb(\backslash mathcal\; A)$ has an $L$-rational point if and only if $\backslash mathcal\; A\; \backslash otimes\_F\; L$ is a symbol algebra. Let $\backslash mathcal\; A$ 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 $C(\backslash mathcal\; A,\; K)$ such that, for a field extension $L/F$ with the property $[KL:L]=[K:F]$, the variety $C(\backslash mathcal\; A,\; K)$ has an $L$-rational point if and only if $KL$ is a subfield of $\backslash mathcal\; A\; \backslash otimes\_F\; L$.

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