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=yn+∑i=0n−1ai(x)yi∈Fq[x][y] with i.i.d. coefficients ai taking values in the set {a(x)∈Fq[x]:deg⁡a≤d} with uniform probability, is irreducible with probability tending to 1−1qd 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 An. Moreover, we prove that if we assume a version of the polynomial Chowla conjecture over Fq[x], then the Galois group of this polynomial is actually equal to the symmetric group Sn with probability tending to 1−1qd. We also study the other possible Galois groups occurring with positive limit probability. Finally, we study the same problems with n fixed and d→∞.