Id\'eaux premiers totalement d\'ecompos\'es et sommes de Newton

TL;DR

This paper develops an effective criterion based on Galois groups and linear recurrence sequences to identify prime ideals in number fields where an irreducible polynomial completely splits, especially for degrees at least 4 with symmetric or alternating Galois groups.

Contribution

It introduces a new criterion linking Galois groups and recurrence sequences to characterize prime splitting in number fields, extending to higher degrees with specific Galois groups.

Findings

Criterion applies to polynomials of degree ≥ 4 with symmetric or alternating Galois groups.

Provides a method to identify prime ideals where the polynomial splits completely.

Connects Galois theory with linear recurrence sequences for prime ideal characterization.

Abstract

Let $K$ be a number field and $f\in K[X]$ an irreducible monic polynomial with coefficients in $O_K$, the ring of integers of $K$. We aim to enounce an effective criterion, in terms of the Galois group of $f$ over $K$ and a linear recurrence sequence associated to $f$, allowing sometimes to characterize the prime ideals of $O_K$ modulo which $f$ completely splits. If $\alpha$ is a root of $f$, this criterion therefore gives a characterization of the prime ideals of $O_K$ which split completely in $K(\alpha)$. It does apply if the degree of $f$ is at least $4$ and the Galois group of $f$ is the symmetric group or the alternating group.