Idéaux premiers totalement décomposés et sommes de Newton

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.