Ostrowski quotients for finite extensions of number fields

Ehsan Shahoseini, Ali Rajaei and Abbas Maarefparvar

Vol. 321 (2022), No. 2, 415–429

DOI: 10.2140/pjm.2022.321.415

Abstract

For $L∕K$, a finite Galois extension of number fields, the relative Pólya group $\mathrm{Po}<!--FUNCTION\; APPLICATION-->(L∕K)$ coincides with the group of strongly ambiguous ideal classes in $L∕K$. In this paper, using a well-known exact sequence related to $\mathrm{Po}<!--FUNCTION\; APPLICATION-->(L∕K)$ from the works of Brumer and Rosen (1963) and Zantema (1982), we find short proofs for some classical results in the literature. Then we define the Ostrowski quotient $\mathrm{Ost}<!--FUNCTION\; APPLICATION-->(L∕K)$ as the cokernel of the capitulation map into $\mathrm{Po}<!--FUNCTION\; APPLICATION-->(L∕K)$ and generalize some known results for $\mathrm{Po}<!--FUNCTION\; APPLICATION-->(L∕\mathbb{Q})$ to $\mathrm{Ost}<!--FUNCTION\; APPLICATION-->(L∕K)$.

Keywords

Pólya group, relative Pólya group, Ostrowski quotient, Galois cohomology, capitulation problem

Mathematical Subject Classification

Primary: 11R29, 11R37, 11R34, 11R21

Milestones

Received: 17 January 2022

Revised: 22 June 2022

Accepted: 26 November 2022

Published: 21 March 2023