  We estimate several probability distributions arising from the study of
random, monic polynomials of degree $n$ with coefficients in the integers of a
general $p$-adic field $K_{\mathfrak{p}}$ having residue field with $q= p^f$
elements. We estimate the distribution of the degrees of irreducible factors of
the polynomials, with tight error bounds valid when $q> n^2+n$. We also
estimate the distribution of Galois groups of such polynomials, showing that
for fixed $n$, almost all Galois groups are cyclic in the limit $q \to \infty$.
In particular, we show that the Galois groups are cyclic with probability at
least $1 - \frac{1}{q}$. We obtain exact formulas in the case of
$K_{\mathfrak{p}}$ for all $p > n$ when $n=2$ and $n=3$.
