The Probability That a Random Monic p-adic Polynomial Splits

Abstract

Let {\small $R$} be a complete discrete valuation ring with finite residue field, and let {\small $r_n$} be the probability that a random monic polynomial over {\small $R$} of degree {\small $n$} factors over {\small $R$} into linear factors. We study {\small $r_n$} in detail. Among other things, we show that {\small $r_n$} satisfies an interesting recursion, make a conjecture on the asymptotic behavior of {\small $r_n$} as {\small $n$} goes to infinity, and prove the conjecture in the case that the residue field has two elements.