Ostrowski quotients for finite extensions of number fields

TL;DR

This paper explores the Ostrowski quotient in finite Galois extensions of number fields, providing new proofs of classical results and generalizing properties of the Pólya group to this new invariant.

Contribution

It introduces the Ostrowski quotient as a new invariant and extends known results about the Pólya group to this broader context.

Findings

Short proofs of classical results using exact sequences.

Definition and analysis of the Ostrowski quotient.

Generalization of properties from $ ext{Po}(L/ ext{Q})$ to $ ext{Ost}(L/K)$.

Abstract

For $L/K$ a finite Galois extension of number fields, the relative P\'olya group $\Po(L/K)$ coincides with the group of strongly ambiguous ideal classes in $L/K$. In this paper, using a well known exact sequence related to $\Po(L/K)$, in the works of Brumer-Rosen and Zantema, we find short proofs for some classical results in the literatur. Then we define the ``Ostrowski quotient'' $\Ost(L/K)$ as the cokernel of the capitulation map into $\Po(L/K)$, and generalize some known results for $\Po(L/\mathbb{Q})$ to $\Ost(L/K)$.