The probability that a p-adic random \'etale algebra is an unramified field

Abstract

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 $1/p$.