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A counter example to Malle’s conjecture on the asymptotics of discriminants. (English) Zbl 1083.11069

Let \(n\) be a natural number, \(G\) a transitive permutation group in \(S_n\) and \(k\) a number field. \(Z(k,G;x)\) denotes the number of extensions \(K/k\) of degree \(n\) such that the normal closure of \(K/k\) has Galois group \(G\) and the absolute norm of the discriminant \(d{K/k}\) is smaller or equal to \(x\). G. Malle [J. Number Theory 92, No. 2, 315–329 (2002; Zbl 1022.11058), Exp. Math. 13, No. 2, 129–135 (2004; Zbl 1099.11065)] has given a precise conjecture about the asymptotic behavior of the function \(Z(k,G;x)\). The author shows that the wreath product of \(C_3\) with \(C_2\) is a counter example for \(k=\mathbb Q\).

MSC:

11R32 Galois theory
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References:

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