The degree of the splitting field of a random polynomial over a finite field

The asymptotics of the order of a random permutation have been widely studied. P. Erdos and P. Turan proved that asymptotically the distribution of the logarithm of the order of an element in the symmetric group S-n is normal with mean (1)(2)(log n)(2) and variance (1)(3)( log n)(3). More recently R. Stong has shown that the mean of the order is asymptotically exp(C rootn/ log n + O(root n log log n/log n)) where C = 2.99047.... We prove similar results for the asymptotics of the degree of the splitting field of a random polynomial of degree n over a finite field.