Symbol algebras and cyclicity of algebras after a scalar extension

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 ρ _n . 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 ).