Bi-quadratic Pólya fields with five distinct ramified primes

For an algebraic number field $K$, the Pólya group of $K$, denoted by $Po(K),$ is the subgroup of the ideal class group $C{l}_K$ generated by the ideal classes of the products of prime ideals of same norm. The number field $K$ is said to be Pólya 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 $K\mathrm{/}\mathbb{Q}$. This extends the previous results on this problem, where the number of distinct ramified primes was at most four.