The S-Relative Pólya Groups and S-Ostrowski Quotients of Number Fields

Let $K/F$ 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 [J Reine Angew Math 613:75–97, 2007], we generalize the notions of the relative Pólya group $\Po(K/F)$ [Chabert in J Number Theory 203:360–375, 2019; Maarefparvar and Rajaei in J Number Theory 207:367-384, 2020] and the Ostrowski quotient $\Ost(K/F)$ [Shahoseini et al. in Pac JMath 321(2):415–429, 2022] 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.