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

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.