Galois groups of random polynomials over the rational function field

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