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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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