Kummer extension and Polya fields

Abstract

A number field is called a Polya field if the module of integer-valued polynomials over its ring of integers has a regular basis. Let a ∈ ℤ, a ≠ 0 and p ≥ 3 be a prime number. In this paper, we characterize the number fields of the form ℚ(ζ_p, (a)^{1∕ p}) which are Polya fields. We hence generalize results of [9]. New bounds for the number of ramified prime numbers in these fields are derived.