Density of $p$-adic polynomials generating extensions with fixed splitting type

Abstract

We prove that the density of polynomials $P(x)={\sum }_{i=0}^n{a}_n{x}^n$ over a local field $K$ generating an étale extension with specified splitting type is a rational function in terms of the size of the residue field of $K$ in the case where the splitting type is tame. Moreover, we give a computable recursive formula for these densities and compute the asymptotics of this density as the size of the residue field tends to infinity.