Geodesic
The degree of the splitting field of a random polynomial over a finite field
The electronic journal of combinatorics, Tome 11 (2004) no. 1
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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 $S_{n}$ is normal with mean ${1\over2}(\log n)^{2}$ and variance ${1\over3}(\log n)^{3}$. More recently R. Stong has shown that the mean of the order is asymptotically $\exp(C\sqrt{n/\log n}+O(\sqrt{n}\log\log n/\log n))$ where $C=2.99047\ldots$. 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.
DOI : 10.37236/1823
Classification : 11T06, 11C08
Mots-clés : finite field, polynomial, degree of splitting field 
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     author = {John D. Dixon and Daniel Panario},
     title = {The degree of the splitting field of a random polynomial over a finite field},
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     number = {1},
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John D. Dixon; Daniel Panario. The degree of the splitting field of a random polynomial over a finite field. The electronic journal of combinatorics, Tome 11 (2004) no. 1. doi: 10.37236/1823

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