Algebraic systems of quadratic forms of number fields and function fields (Q1264171)

If \(F\subset L\) is a finite extension of fields of characteristic \(\neq 2\) then a quadratic form \(\phi\) over \(F\) is algebraic (resp. a scaled trace form) if \(\phi\) is Witt equivalent to the trace form \(\text{Tr}_{L/F}(<1>)\) (resp. to \(\text{Tr}_{L/D}(<\lambda >)\) for some \(\lambda \in L^*)\). Given a form \(\phi\) over \(F\) one wants to decide if \(\phi\) is algebraic. For some classes of fields (e.g. number fields, function fields in one variable over a real closed field) a simple criterion is known. Essentially the question is raised for systems of forms: If \(\phi_1,\dots,\phi_n\) are forms over \(F\), when do there exist a finite field extension \(F\subset L\) and \(\lambda_1=1\), \(\lambda_2,\dots,\lambda_n\in L^*\) such that \(\phi_i\) is Witt equivalent to \(\text{Tr}_{L/F}(<\lambda_i>)?\) An answer is provided for the classes of fields mentioned above.