First-degree prime ideals of composite extensions

TL;DR

This paper investigates the relationship between first-degree prime ideals in composite number fields and their constituent fields, providing a classification of exceptions and demonstrating computational efficiencies in prime ideal calculations.

Contribution

It establishes a near-universal correspondence between prime ideals of composite and component fields, classifies exceptions, and shows computational improvements in prime ideal determination.

Findings

Almost always, prime ideals in composite fields relate to those in component fields.

Explicit counterexamples where the correspondence fails are provided.

Computational methods are improved, reducing time for prime ideal calculations.

Abstract

Let $\mathbb{Q}(\alpha)$ and $\mathbb{Q}(\beta)$ be linearly disjoint number fields and let $\mathbb{Q}(\theta)$ be their compositum. We prove that the first-degree prime ideals of $\mathbb{Z}[\theta]$ may almost always be constructed in terms of the first-degree prime ideals of $\mathbb{Z}[\alpha]$ and $\mathbb{Z}[\beta]$, and vice-versa. We also classify the cases in which this correspondence does not hold, by providing explicit counterexamples. We show that for every pair of coprime integers $d,e \in \mathbb{Z}$, such a correspondence almost always respects the divisibility of principal ideals of the form $(e+d\theta)\mathbb{Z}[\theta]$, with a few exceptions that we characterize. Finally, we discuss the computational improvement of such an approach, and we verify the reduction in time needed for computing such primes for certain concrete cases.