Ulf Rehmann, Sergey V. Tikhonov, and Vyacheslav I. Yanchevskii: Symbols and cyclicity of algebras after a scalar extension

rehmann@math.uni-bielefeld.de, tsv@im.bas-net.by, yanch@im.bas-net.by

Submission: 2008, Dec 1

1. 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 $E$.
2. Let $\cA$ 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 $\Symb(\cA)$ such that, for a field extension $L/F$, the variety $\Symb(\cA)$ has an $L$-rational point iff $\cA \otimes_F L$ is a symbol algebra.
3. Let $\cA$ be a central simple algebra over a field $F$ of degree $n$ and $K/F$ a cyclic field extension of degree $n$. We construct a quasi-affine $F$-variety $C(\cA, K)$ such that, for a field extension $L/F$ with the property $[K L:L]=[K:F]$ ,the variety $C(\cA, K)$ has an $L$-rational point iff $K L$ is a subfield of $\cA \otimes_F L$.

2000 Mathematics Subject Classification: 16K50

Keywords and Phrases: central simple algebras, Brauer groups, splitting varieties

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