@article{859760766b0a4b409d8b5d55b8fe5464,

title = "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]:dega≤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 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 [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→∞.",

keywords = "Finite fields, Galois theory, Polynomials, Probability",

author = "Alexei Entin and Alexander Popov",

note = "Publisher Copyright: {\textcopyright} 2024 Elsevier Inc.",

year = "2024",

month = sep,

doi = "10.1016/j.ffa.2024.102466",

language = "אנגלית",

volume = "98",

journal = "Finite Fields and Their Applications",

issn = "1071-5797",

publisher = "Academic Press Inc.",

}