  We study the random \'etale algebra generated by a random polynomial with
i.i.d. coefficients distributed according to Haar measure normalized on
$\mathbb{Z}_p$. We determine the probability that this random algebra is an
unramified field, explicitly. In addition, we prove a private case of a
conjecture made by Bhargava, Cremona, Fisher, and Gajovi\'c. More precisely, we
show that this probability is a rational function of $p$ that is invariant
under replacing $p$ by $1/p$.
