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

Malle, Gunter

Exp. Math. 13, No. 2, 129-135 (2004).

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.

Reviewer: Christian U. Jensen (København)

MSC:

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

Keywords:

density of extensions; distribution of discriminants

Citations:

Zbl 1022.11058

Software:

KANT/KASH

References:

[1]	Bhargava M., PhD. diss., in: ”Higher Composition Laws.” (2001) · Zbl 1129.11049
[2]	Cohen H., A Course in Computational Algebraic Number Theory. (1993) · Zbl 0786.11071
[3]	DOI: 10.1007/978-1-4419-8489-0 · Zbl 0977.11056 · doi:10.1007/978-1-4419-8489-0
[4]	Cohen H., Counting Discriminants of Number Fields. (2000) · Zbl 0987.11080
[5]	DOI: 10.1007/3-540-45455-1_7 · doi:10.1007/3-540-45455-1_7
[6]	DOI: 10.1023/A:1016310902973 · Zbl 1050.11104 · doi:10.1023/A:1016310902973
[7]	DOI: 10.1006/jsco.1996.0126 · Zbl 0886.11070 · doi:10.1006/jsco.1996.0126
[8]	Klüners J., Journal reine angew. Math.
[9]	Koch H., Number Theory–Algebraic Numbers and Functions (2000)
[10]	DOI: 10.1006/jnth.2001.2713 · Zbl 1022.11058 · doi:10.1006/jnth.2001.2713
[11]	DOI: 10.1112/plms/s3-58.1.17 · Zbl 0628.12006 · doi:10.1112/plms/s3-58.1.17