Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

First-degree prime ideals of composite extensions

Giordano Santilli, Daniele Taufer

TL;DR

The paper addresses how first-degree prime ideals in a composite extension $ abla( heta)$, with $ abla( heta)$ the compositum of linearly disjoint fields $ abla( ext{Q})( abla)$, can be read from and constructed from the corresponding primes in the subfields $ abla( ext{Q})[ abla]$ and $ abla( ext{Q})[eta]$. It develops a precise combination rule $(r,s) o(r+s,p)$ for first-degree primes via the resultant $R_{f,g}$ and proves this correspondence holds almost always, with a complete classification of exceptional cases tied to roots of an affine map modulo $p$; it also analyzes how principal ideals $(e+d heta)$ factor through the subfields via explicit norms $ abla_ abla$ and $ abla_eta$. The paper further demonstrates significant computational gains by solving the prime-ideal problem in the smaller subfields and composing the results, with detailed asymptotic complexity and empirical studies for degree-6 and smooth-degree extensions. Overall, these results provide both theoretical insight into the structure of primes in composite fields and practical methods to accelerate GNFS-like computations that rely on first-degree prime bases.

Abstract

Let $\mathbb{Q}(α)$ and $\mathbb{Q}(β)$ be linearly disjoint number fields and let $\mathbb{Q}(θ)$ be their compositum. We prove that the first-degree prime ideals of $\mathbb{Z}[θ]$ may almost always be constructed in terms of the first-degree prime ideals of $\mathbb{Z}[α]$ and $\mathbb{Z}[β]$, 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 $(e+dθ)\mathbb{Z}[θ]$, 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.

First-degree prime ideals of composite extensions

TL;DR

The paper addresses how first-degree prime ideals in a composite extension , with the compositum of linearly disjoint fields , can be read from and constructed from the corresponding primes in the subfields and . It develops a precise combination rule for first-degree primes via the resultant and proves this correspondence holds almost always, with a complete classification of exceptional cases tied to roots of an affine map modulo ; it also analyzes how principal ideals factor through the subfields via explicit norms and . The paper further demonstrates significant computational gains by solving the prime-ideal problem in the smaller subfields and composing the results, with detailed asymptotic complexity and empirical studies for degree-6 and smooth-degree extensions. Overall, these results provide both theoretical insight into the structure of primes in composite fields and practical methods to accelerate GNFS-like computations that rely on first-degree prime bases.

Abstract

Let and be linearly disjoint number fields and let be their compositum. We prove that the first-degree prime ideals of may almost always be constructed in terms of the first-degree prime ideals of and , 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 , such a correspondence almost always respects the divisibility of principal ideals of the form , 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.

Paper Structure

This paper contains 11 sections, 19 theorems, 51 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Resultant
  4. Linearly disjoint extensions
  5. First-Degree Prime Ideals
  6. Divisibility of prescribed principal ideals
  7. Computational improvement
  8. Asymptotic complexity
  9. Degree-6 extensions
  10. Extensions of smooth degrees
  11. Conclusions

Key Result

Proposition 2.2

Let $f,g \in \mathbb{k}[x]$ as above, and let $L$ be an extension of $\mathbb{k}$ where both $f$ and $g$ split completely, i.e. Then

Figures (3)

  • Figure 1: Time needed to compute first-degree prime ideals of norm up to $M$ for a degree-$6$ defining polynomial.
  • Figure 2: Lattice of the minimal fields in a number field of degree $315$. The large extension is realized as the compositum of the small underlying fields.
  • Figure 3: Time needed to compute first-degree prime ideals of norm up to $M$ for a degree-$315$ defining polynomial.

Theorems & Definitions (41)

  • Definition 2.1: Resultant
  • Proposition 2.2: Lang
  • Corollary 2.3: Lang
  • Corollary 2.4: Lang
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Proposition 2.8: Cohn
  • Definition 2.9: Linearly disjointness
  • Lemma 2.10
  • ...and 31 more