On dihedral Pólya fields

A number field is a Pólya field when the module of integer-valued polynomials over its ring of integers has a regular basis. A quartic field is a $D_4$-field when the Galois group of its splitting field is the dihedral group $D_4$ of 8 elements. In this paper, we prove that there are infinitely many $D_4$-Pólya fields with ℓ ramified prime numbers for each $\ell \in \{2,3,4,5\}$ and a $D_4$ Pólya field with $\ell =1$ ramified prime number, is, up to $Q$-isomorphism, $Q\left(\sqrt{1+\sqrt{2}}\right)$, $Q\left(\sqrt{-1+\sqrt{2}}\right)$ or a pure field. Consequently, we answer a question raised in [29] on $D_4$-fields. The same question arises on pure fields. We find an upper bound for such fields. And for any integer ℓ less that this bound, we show that there are infinitely many pure Pólya fields with ℓ ramified prime numbers except when $\ell =1$ where we proved that there are only 2 fields (and their two conjugate fields)