Probabilistic Galois Theory in Function Fields

TL;DR

This paper investigates the irreducibility and Galois groups of random polynomials over function fields, establishing probabilistic results and conditions under which the Galois group is the full symmetric group.

Contribution

It provides new probabilistic results on the irreducibility and Galois groups of random polynomials over function fields, including conditions for the Galois group to be the full symmetric group.

Findings

Probability of irreducibility tends to 1 - 1/q^d as n→∞.

Galois group contains the alternating group A_n with high probability.

Under a polynomial Chowla conjecture, Galois group equals S_n with high probability.

Abstract

We study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial $f=y^n+\sum_{i=0}^{n-1}a_i(x)y^i\in\mathbb F_q[x][y]$ with i.i.d coefficients $a_i$ taking values in the set $\{a(x)\in\mathbb{F}_q[x]: \mathrm{deg}\, a\leq d\}$ with uniform probability, is irreducible with probability tending to $1-\frac{1}{q^d}$ as $n\to\infty$, where $d$ and $q$ are fixed. We also prove that with the same probability, the Galois group of this random polynomial contains the alternating group $A_n$. Moreover, we prove that if we assume a version of the polynomial Chowla conjecture over $\mathbb{F}_q[x]$, then the Galois group of this polynomial is actually equal to the symmetric group $S_n$ with probability tending to $1-\frac{1}{q^d}$. We also study the other possible Galois groups occurring with positive limit probability. Finally, we study the same problems with $n$ fixed and $d\to\infty$.