On trace forms of algebraic function fields.

*(English)*Zbl 0702.11021The 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.

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.

Reviewer: Kazimierz Szymiczek (Katowice)

##### 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 |