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Let $K$ be a number field and $f\in K[X]$ an irreducible monic polynomialwith coefficients in $O_K$, the ring of integers of $K$. We aim to enounce aneffective criterion, in terms of the Galois group of $f$ over $K$ and a linearrecurrence sequence associated to $f$, allowing sometimes to characterize theprime ideals of $O_K$ modulo which $f$ completely splits. If $\alpha$ is a rootof $f$, this criterion therefore gives a characterization of the prime idealsof $O_K$ 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 thealternating group.

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