# zbMATH — the first resource for mathematics

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
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.

##### MSC:
 11R29 Class numbers, class groups, discriminants 11R16 Cubic and quartic extensions
##### Keywords:
discriminants; density; cyclic extensions; Kummer theory
Full Text: