The Proportion of Irreducible p-adic Polynomials

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 ≠ n, and also when $n = 4$ and p ≠ 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.