The Ostrowski Quotient for a finite extension of number fields

For a number field $K$, the P\'olya group of $K$, denoted by $Po(K)$, is the subgroup of the ideal class group of $K$ generated by the classes of the products of maximal ideals of $K$ with the same norm. In this talk, after reviewing some results concerning $Po(K)$, I will generalize this notion to the relative P\'olya group $Po(K/F)$, for $K/F$ a finite extension of number fields. Accordingly, I will generalize some results in the literature about P\'olya groups to the relative case. Then, due to some essential observations, I will explain why we need to modify the notion of the relative P\'olya group to the Ostrowski quotient $Ost(K/F)$ to get a more 'accurate' generalization of $Po(K)$. The talk is based on a joint work with Ali Rajaei (Tarbiat Modares University) and Ehsan Shahoseini (Institute For Research In Fundamental Sciences).