Density of discriminants of cyclic extensions of prime degree. (Densité des discriminants des extensions cycliques de degré premier.) (French. Abridged English version) Zbl 0941.11042

Cohen, Henri; Diaz y Diaz, Francisco; Olivier, Michel

C. R. Acad. Sci., Paris, Sér. I, Math. 330, No. 2, 61-66 (2000).

Let \(K\) be a number field and \(G\) a finite abelian group. Let \(N_K(G,x)\) denote the number of \(K\)-isomorphism classes of extensions \(L/K\) with Galois group \(G\) whose relative discriminant has absolute norm \(\leq x\). It was shown by D. J. Wright [Proc. Lond. Math. Soc. (3) 58, 17-50 (1989; Zbl 0628.12006)] that there are constants \(c_K(G)\), \(\alpha\) and \(\beta\) such that \(N_K(G,x) \sim c_K(G) x^\alpha (\log x)^\beta\), and Wright also gave explicit formulas for \(\alpha\) and \(\beta\) in terms of invariants of \(G\) and \(K\). In this paper, the authors provide explicit formulas for the constants \(c_K(G)\) when \(G\) is cyclic of prime degree. The proof, which is only sketched and whose details will be published elsewhere, depends on Kummer theory and class field theory, and involves delicate technical calculations.

Reviewer: Franz Lemmermeyer (Heidelberg)

MSC:

11R29	Class numbers, class groups, discriminants
11R16	Cubic and quartic extensions

Keywords:

discriminants; density; cyclic extensions; Kummer theory

Citations:

Zbl 0628.12006