Some non-Pólya biquadratic fields with low ramification

Abstract

Pólya fields are fields with principal Bhargava factorial ideals, and as a generalization of class number one number fields, their classification might be of interest to number theorists. It is known that Pólya fields have little ramification, and the aim of this paper is to prove non-Pólyaness of an infinite family of biquadratic number fields with 3 or 4 primes of ramification, correcting a minor mistake in the literature. It turns out that finer arithmetic invariants of the field such as the Hasse unit index plays a direct role in some cases.