On trace forms of algebraic function fields
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. \textit{P. E. Conner} and \textit{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 \textit{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.
scientific article; zbMATH DE number 4150274