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 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√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.

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Journal Electronic Journal of Combinatorics
Citation
Dixon, J.D. (John D.), & Panario, D. (2004). The degree of the splitting field of a random polynomial over a finite field. Electronic Journal of Combinatorics, 11(1 R), 1–10.