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We study the distribution of the roots of a random p-adic polynomial in an algebraic closure of
${\mathbb Q}_p$
. We prove that the mean number of roots generating a fixed finite extension K of
${\mathbb Q}_p$
depends mostly on the discriminant of K, an extension containing fewer roots when it becomes more ramified. We prove further that for any positive integer r, a random p-adic polynomial of sufficiently large degree has about r roots on average in extensions of degree at most r.
Beyond the mean, we also study higher moments and correlations between the number of roots in two given subsets of
${\mathbb Q}_p$
(or, more generally, of a finite extension of
${\mathbb Q}_p$
). In this perspective, we notably establish results highlighting that the roots tend to repel each other and quantify this phenomenon.
