The Probability That a Random Monic p-adic Polynomial Splits

Abstract

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

Keywords:

• Primary 11S05
• p-adic
• polynomial
• splits completely