Probabilistic Galois theory in function fields

We study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial f=yⁿ+∑_i=0ⁿ⁻¹a_i(x)yⁱ∈F_q[x][y] with i.i.d. coefficients a_i taking values in the set {a(x)∈F_q[x]:deg⁡a≤d} with uniform probability, is irreducible with probability tending to [Formula presented] as n→∞, 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[x], then the Galois group of this polynomial is actually equal to the symmetric group S_n with probability tending to [Formula presented]. We also study the other possible Galois groups occurring with positive limit probability. Finally, we study the same problems with n fixed and d→∞.