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 $f=y^n+{\sum }_{i=0}^{n-1}{a}_i(x)y^i\in {\mathbb{F}}_q[x][y]$ with i.i.d coefficients $a_i$ taking values in the set $\{a(x)\in {\mathbb{F}}_q[x]:\mathrm{deg}a\le d\}$ 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 ${\mathbb{F}}_q[x]$, 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 $d\to \infty$.

v2: changed conditional results to depend on a more standard version of Chowla's conjecture. Added references acknowledgments. Fixed small errors, typos and style issues. v3: fixed a few small errors