Galois groups of random polynomials over the rational function field

Abstract For a fixed prime power and natural number , we consider a random polynomial with drawn uniformly and independently at random from the set of all polynomials in of degree . We show that with probability tending to 1 as the Galois group of over is isomorphic to , where is cyclic, and are small quantities with a simple explicit dependence on . As a corollary we deduce that as . Thus, we are able to overcome the versus ambiguity in the most natural restricted coefficients random polynomial model over , which has not been achieved over so far.