Probabilistic Galois theory in function fields

Abstract

We study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial $f=y^n+{\sum }_{i=0}^{n-1}{a}_i\left(x\right)y^i\in F_q\left[x\right]\left[y\right]$ with i.i.d. coefficients $a_i$ taking values in the set $\left\{a\right(x)\in F_q[x]:\mathrm{deg}a\le d\}$ 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 $F_q\left[x\right]$, 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 $d\to \infty$.