Powers of totally positive subgroups of fields (Q958519)

A subgroup \(S\) of the multiplicative group \(F^\times\) of a field \(F\) is said to be co-torsion if the quotient group \(F^\times/S\) is a torsion group, and is said to be totally positive if \(S \subseteq \sum (F^\times)^2\). The main result of the paper states that if \(F\) is a formally real field and \(m\) is any fixed positive integer \(m\), then for any family \(S_\lambda\), \(\lambda \in \Lambda\), of totally positive co-torsion subgroups of \(F^\times\) one has \[ E^+ \cdot (\sum_{\lambda \in \Lambda} \sum S_\lambda)^m = \sum_{\lambda \in \Lambda} \sum S_\lambda^m, \] where \(E^+\) is the set of totally positive invertible elements of the real holomorphy ring of \(F\). This generalizes a result of [\textit{E. Becker}, The real holomorphy ring and sums of \(2n\)-th powers. Géométrie algébrique réelle et formes quadratiques, Journées Soc. Math. Fr., Univ. Rennes 1981, Lect. Notes Math. 959, 139--181 (1982; Zbl 0508.12020)] who considered the case where the family of subgroups \(S_\lambda\) is replaced with the subgroup \(S = (F^\times)^n\), \(n\) even. As a consequence one obtains some finiteness statements about the \(S\)-Pythagoras number of \(F\).