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

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 \(\ell\).

MSC:

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