First-Degree Prime Ideals of Biquadratic Fields Dividing Prescribed Principal Ideals

Abstract

We describe first-degree prime ideals of biquadratic extensions in terms of the first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms. The correspondence between these ideals in the larger ring and those in the smaller ones extends to the divisibility of specially-shaped principal ideals in their respective rings, with some exceptions that we explicitly characterize.

Keywords

• first-degree prime ideals
• biquadratic extensions
• ideal division