The $S$-relative Pólya groups and $S$-Ostrowski quotients of number fields

Let $K\mathrm{/}F$ be a finite extension of number fields and $S$ be a finite set of primes of $F$, including all the archimedean ones. In this paper, using some results of González-Avilés \cite{Aviles}, we generalize the notions of the relative Pólya group $\text{Po}(K\mathrm{/}F)$ \cite{ChabertI,MR2} and the Ostrowski quotient $\text{Ost}(K\mathrm{/}F)$ \cite{SRM} to their $S$-versions. Using this approach, we obtain generalizations of some well-known results on the $S$-capitulation map, including an $S$-version of Hilbert’s theorem 94.