## On the distribution of Galois groups. II.(English)Zbl 1099.11065

This paper is a continuation of the author’s earlier paper [J. Number Theory 92, No. 2, 315–329 (2002; Zbl 1022.11058)]. Let $$k$$ be a number field and $$G$$ a transitive subgroup of the symmetric group $$S_n$$. Let $$K/k$$ be a finite extension such that the Galois group of the Galois closure $$K/I$$ as a permutation group is isomorphic to $$G$$. The paper concerns the asymptotic behaviour $$Z(k,G;x)$$ defined as the number of field extensions of $$k$$ of degree $$n$$ with Galois group isomorphic to $$G$$ and with the norm of the relative discriminant $$d_{K/k}$$ bounded by $$x$$. The author proposes the conjecture that $$Z(k,G;x)$$ is asymptotically equal to $$cx^{a(G)}(\log x)^{b(k,G)-1}$$, where $$c>0$$, $$0<a(G)\leq 1$$ and $$b(K,g)\in\mathbb N$$ are constants. Very precise values are proposed for the constants $$a(G)$$ and $$b(k,G)$$, $$a(G$$ only depending on $$G$$ and $$b(k,G)$$ depending on $$G$$ and the absolute Galois group of $$k$$. The conjecture is known to be true for abelian groups and some non-abelian groups of small order (e.g., $$A_4$$, $$S_3$$, and $$D_4)$$. Finally some computational data are provided to add to the numerical evidence of the conjecture for the nonsolvable groups of degree 5.

### MSC:

 11R32 Galois theory 11R47 Other analytic theory 12F10 Separable extensions, Galois theory

### Keywords:

density of extensions; distribution of discriminants

Zbl 1022.11058

KANT/KASH
Full Text:

### References:

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