Probabilistic Galois Theory in Function Fields

2023/11/24 by Entin, Alexei, Popov, Alexander
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2311.14862

Abstract

We study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial f=yⁿ+∑_i=0^n-1a_i(x)yⁱ∈\mathbb F_q[x][y] with i.i.d coefficients a_i taking values in the set \a(x)∈\mathbbF_q[x]: deg a≤ d\ with uniform probability, is irreducible with probability tending to 1-(1)/(q^d) 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 \mathbbF_q[x], then the Galois group of this polynomial is actually equal to the symmetric group S_n with probability tending to 1-(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→∞.

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