Probabilistic Galois Theory over $P$-adic Fields

TL;DR

This paper investigates the probability distributions of polynomial factorizations and Galois groups over $p$-adic fields, revealing that most Galois groups tend to be cyclic as the residue field size grows.

Contribution

It provides new probabilistic estimates for polynomial factorizations and Galois groups over $p$-adic fields, including exact formulas for small degrees.

Findings

Most Galois groups are cyclic as $q o fty$

Probability of cyclic Galois group is at least $1 - 1/q$

Exact formulas obtained for degrees 2 and 3 when $p > n$

Abstract

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 $n=3$.