About the embedding of a number field in a Pólya field

One knows the classical problem of the embedding of a number field K in a field with class number one. This problem has a negative answer. In this article, we consider a new embedding problem: Is every number field contained in a Pólya field? A Pólya field is a number field K   such that all the characteristic ideals In(K)In(K) are principal. We give a positive answer to this problem: the Hilbert class field HKHK of K   is a Pólya field. However, HKHK is not necessarily the smallest Pólya field containing K. Thus, we give upper bounds for the Pólya number of K, namely the minimal degree of a Pólya field containing K.