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+ n-1i=0ai(x)yi. Fq[x][y] with i.i.d. coefficients aitaking values in the set {a(x). Fq[x] : dega = d} with uniform probability, is irreducible with probability tending to 1 - 1qdas n.8, where dand qare 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 Snwith 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 nfixed and d.8. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.