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Let $K/F$ be a finite extension of number fields and $S$ be a finite set ofprimes of $F$, including all the archimedean ones. In this paper, using someresults of Gonz\'alez-Avil\'es \cite{Aviles}, we generalize the notions of therelative P\'olya group $\Po(K/F)$ \cite{ChabertI,MR2} and the Ostrowskiquotient $\Ost(K/F)$ \cite{SRM} to their $S$-versions. Using this approach, weobtain generalizations of some well-known results on the $S$-capitulation map,including an $S$-version of Hilbert's theorem 94.

The $S$-relative P\'olya groups and $S$-Ostrowski quotients of number fields