Symbol algebras and cyclicity of algebras after a scalar extension

Abstract

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 nth root of unity ρn. We construct a quasi-affine F-variety Symb(\( \mathcal{A} \)) such that for a field extension L/F Symb(\( \mathcal{A} \)) 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(\( \mathcal{A} \),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 \( \mathcal{A}{ \otimes_F}L \).