  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 $d\to\infty$.
