## 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

### Citations:

Zbl 1022.11058; Zbl 1099.11065
Full Text:

### References:

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