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On trace forms of algebraic function fields. (English) Zbl 0702.11021
The trace functional of a finite separable field extension $$L/K$$ induces a bilinear symmetric form on $$L$$ with values in $$K$$ denoted here $$T_K(L,1)$$. If $$K$$ is formally real field, then the signature $$\operatorname{sgn}_P T_K(L,1)$$ counts the number of extensions of the ordering $$P$$ of $$K$$ to $$L$$, hence is always nonnegative. Also the forms Witt equivalent to $$T_K(L,1)$$ have all signatures nonnegative.
P. E. Conner and R. Perlis [A survey of trace forms of algebraic number fields (1984; Zbl 0551.10017)] asked if every regular form over $$K$$ with all signatures nonnegative is necessarily Witt equivalent to the trace form of a finite extension $$L/K$$. They proved this is so when $$K= \mathbb Q$$, and W. Scharlau [Math. Z. 196, 125–127 (1987; Zbl 0658.10025)] extended the result to $$K$$ any algebraic number field. The present author gives an affirmative answer in the case when $$K$$ is an algebraic function field in one variable over a real closed field.
While the general plan of the proof is the same as in Scharlau’s paper (reduction to one-dimensional forms and then a proof in this case), a new approach is required to deal with the infinite number of orderings of $$K$$. The approach is based on the fact that $$K$$ allows “effective diagonalization” of quadratic forms.

##### MSC:
 11E10 Forms over real fields 11E12 Quadratic forms over global rings and fields 11E81 Algebraic theory of quadratic forms; Witt groups and rings 11R58 Arithmetic theory of algebraic function fields
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