Bi-quadratic P\'{o}lya fields with five distinct ramified primes

TL;DR

This paper investigates the Pólya group of bi-quadratic number fields with five ramified primes, extending previous work that considered up to four ramified primes, to understand their algebraic structure.

Contribution

It extends the analysis of Pólya groups to bi-quadratic fields with five ramified primes, providing new insights into their algebraic properties.

Findings

Characterization of Pólya groups in fields with five ramified primes

Extension of previous results from four to five ramified primes

Identification of conditions for trivial Pólya groups in these fields

Abstract

For an algebraic number field $K$, the P\'{o}lya group of $K$, denoted by $Po(K),$ is the subgroup of the ideal class group $Cl_{K}$ generated by the ideal classes of the products of prime ideals of same norm. The number field $K$ is said to be P\'{o}lya if $Po(K)$ is trivial. Motivated by several recent studies on the group $Po(K)$ when $K$ is a totally real bi-quadratic field, we investigate the same with five distinct odd primes ramifying in $K/\mathbb{Q}$. This extends the previous results on this problem, where the number of distinct ramified primes was at most four.