On the density of discriminants of cyclic extensions of prime degree. (English) Zbl 1004.11063

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

J. Reine Angew. Math. 550, 169-209 (2002).

Let \(K\) be a number field and \(G\) an abelian group of order \(n\). Let \(N_K(G,x)\) denote the number of extensions \(L/K\) with Galois group \(G\) and relative discriminant \(\leq x\). It is known [M. J. Taylor, J. Lond. Math. Soc. (2) 29, 211-223 (1984; Zbl 0535.12009) and D. J. Wright, Proc. Lond. Math. Soc. (3) 58, 17-50 (1989; Zbl 0628.12006)] that there exist constants \(a(G)\), \(b_K(G)\), and \(c_K(G) > 0\) such that \[ N_K(G,x) \sim c_K(G) x^{a(G)} (\log x)^{b_K(G)}, \] where \(a(G)\) and \(b_K(G)\) can easily be computed from \(G\) and \(K\).
In this article, the authors compute \(c_K(G)\) for the groups \(G = \mathbb Z/\ell\mathbb Z\) (where \(\ell\) is an odd prime) and arbitrary number fields \(K\). The proof uses Kummer theory; first, the cyclotomic extensions \(K(\zeta_\ell)/K\) have to be studied in detail (for the proofs, the reader is referred to the authors’ preprint ‘Cyclotomic extensions of number fields’), then follow calculations of the conductors of the Kummer extensions, and finally the Kummer correspondence is used to count the cyclic extensions of degree \(\ell\).

Reviewer: Franz Lemmermeyer (San Marcos)

MSC:

11R29	Class numbers, class groups, discriminants
11R20	Other abelian and metabelian extensions

Keywords:

discriminants; density; Kummer extensions; cyclotomic extensions; cyclic extensions; conductor

Citations:

Zbl 0535.12009; Zbl 0628.12006

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