The Proportion of Irreducible p-adic Polynomials

TL;DR

This paper calculates the exact proportion of monic p-adic polynomials that are irreducible for specific degrees and primes, providing rational function formulas and extending previous work on polynomial root counts.

Contribution

It provides exact formulas for the irreducibility proportion of monic p-adic polynomials when degree n is prime and p ≠ n, and for degree 4 when p ≠ 2.

Findings

Exact rational function formulas for prime degree n

Exact formula for degree 4 when p ≠ 2

Extension of previous work on polynomial root distributions

Abstract

We attempt to quantify the exact proportion of monic $p$-adic polynomials of degree $n$ which are irreducible. We find an exact answer to this when $n$ is prime and $p \neq n$, and also when $n = 4$ and $p \neq 2$. Our answers are rational functions in $p$. This relates to previous work done to find exact proportions of $p$-adic polynomials of degree $n$ which have $k$ roots.