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 \({{\,\textrm{Po}\,}}(K/F)\) $\text{Po}(K/F)$ (Chabert in J Number Theory 203:360–375, 2019; Maarefparvar and Rajaei in J Number Theory 207:367-384, 2020) and the Ostrowski quotient \({{\,\textrm{Ost}\,}}(K/F)\) $\text{Ost}(K/F)$ (Shahoseini et al. in Pac J Math 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.