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Algebraic systems of quadratic forms of number fields and function fields. (English) Zbl 0689.10027
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.

MSC:
 11E04 Quadratic forms over general fields 11E16 General binary quadratic forms 11R58 Arithmetic theory of algebraic function fields
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References:
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