Symbols and Cyclicity of Algebras after a Scalar Extension
نویسندگان
چکیده
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 A be a central simple algebra over a field F of degree n with a primitive n-th root of unity ρn. We construct a quasiaffine F -variety Symb(A) such that, for a field extension L/F , the variety Symb(A) has an L-rational point iff A ⊗F L is a symbol algebra. 3. Let A 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(A,K) such that, for a field extension L/F with the property [KL : L] = [K : F ] ,the variety C(A,K) has an L-rational point iff KL is a subfield of A⊗F L.
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