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Statistiques sur \(\mathbb F_ q[X]\). (French) Zbl 0519.60006

Let \(P\) be a univariate polynomial of degree \(n\) over a finite field \(K\). We study the degree \(r\) of the splitting field of \(P\) over \(K\). We show that \(\text{Log}(r)\) is “almost always” close to \(\tfrac12\text{Log}^2n\). The method of proof is related to previous work of P. Erdős and P. Turán [Z. Wahrscheinlichkeitstheor. Verw. Geb. 4, 175–186 (1965; Zbl 0137.25602)] in statistical group theory. The results of our paper have been improved by the second author who showed that \(\text{Log}(r)\) looks like a Gaussian random variable.
Reviewer: M. Mignotte

MSC:

11T06 Polynomials over finite fields
11N45 Asymptotic results on counting functions for algebraic and topological structures

Citations:

Zbl 0137.25602
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References:

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