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

TL;DR

This paper calculates the probability that a randomly generated p-adic étale algebra is an unramified field, providing explicit formulas and confirming a special case of a conjecture related to p-adic fields.

Contribution

It explicitly determines the probability of unramified fields among p-adic étale algebras and proves a specific case of a conjecture by Bhargava et al.

Findings

Probability is a rational function of p invariant under p to 1/p

Explicit formula for the probability of unramified fields

Confirmed a special case of a conjecture by Bhargava, Cremona, Fisher, and Gajović

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 $1/p$.