# First-Degree Prime Ideals of Biquadratic Fields dividing prescribed Principal Ideals

## TL;DR

This paper characterizes first-degree prime ideals in biquadratic fields using quadratic subfields, providing explicit conditions for their identification and divisibility, with potential applications in algebraic number theory.

## Contribution

It introduces a novel description of prime ideals in biquadratic fields based on quadratic subfields, extending divisibility relations and offering explicit numerical criteria.

## Key findings

- Prime ideals in biquadratic fields are characterized via quadratic subfields.
- Explicit numerical conditions determine prime ideal divisibility.
- The correspondence extends to principal ideals with specific exceptions.

## Abstract

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