A note on Polya groups

TL;DR

This paper investigates the structure of Pólya groups in number fields, proving that any finite abelian group can be realized as a Pólya group of some number field, thus expanding understanding of their possible configurations.

Contribution

It demonstrates that every finite abelian group can be realized as a Pólya group of a number field, establishing a broad class of possible Pólya group structures.

Findings

Every finite abelian group is isomorphic to a Pólya group of some number field.

The Pólya group is generated by classes of ambiguous ideals in Galois extensions.

The paper provides new insights into the structure and realizability of Pólya groups.

Abstract

Let $K/\mathbf{Q}$ be a finite Galois extension. The P\'olya group of $K$ is the subgroup of the class group $Cl(K)$, generated by the classes of ambiguous ideals of $K$. In this note, among other results, we prove that every finite abelian group is isomorphic to the P\'olya group of a number field.