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. Erdös and P. Turán proved that asymptotically the distribution of the logarithm of the order of an element in the symmetric group Sn is normal with mean 12(log n) 2 and variance 13(log n) 3. More recently R. Stong has shown that the mean of the order is asymptotically exp(C √ n/ log n + O( √ 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.