The degree of the splitting field of a random polynomial over a finite field (Q1773184)

Assuming that all monic polynomials of degree \(n\) over a finite field of \(q\) elements are equally likely [\textit{J.-L. Nicolas}, Topics in Classical Number Theory, Vol. I, II (Budapest, 1981), Colloq. Math. Soc. János Bolyai 34, 1127--1162 (1984; Zbl 0548.10034)] proved that the logarithmic degree of the splitting field of a random polynomial is asymptotically normally distributed with mean asymptotic to \(\frac12(\log n)^2\) and variance asymptotic to \(\frac13(\log n)^3\), following the paper by \textit{P. Erdős} and \textit{P. Turán} [Acta Math. Acad. Sci. Hung. 18, 309--320 1967; Zbl 0235.20003)] for the logarithmic order of random permutations. The authors of this paper show that the expected degree \(E_n(q)\) of the splitting field of a random polynomial satisfies \[ \log E_n(q) = C\sqrt{\frac{n}{\log n}}\left( 1+O\left(\frac{\log \log n}{\sqrt{\log n}}\right)\right), \] where \[ C = 2\sqrt{2\int_0^\infty \frac{\log(1+x)}{e^x-1}\,dx} \approx 2.99047\dots, \] following \textit{R.\ Stong} [Electron. J. Comb. 5, Research paper R41 (1998; Zbl 0907.11031)] for the expected order of random permutations.