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On the equidistribution of Frobenius in cyclic extensions of a number field. (English) Zbl 0535.12009
For a fixed number field K and a positive integer \(n>1\) the author firstly considers the number \(\tilde N\)(x) of all cyclic extensions L/K of degree n such that the conductor of a generating character of \(Hom(Gal(L/K),{\mathbb{C}}^{\times})\) has absolute norm \(f_{L/K}\leq x.\) In the whole article one makes the technical assumption that K contains the \(2^ g\)-th roots of unity for \(g=ord_ 2 n\). The first result gives the asymptotic behaviour of \(\tilde N\)(x) by the formula \[ \lim_{x\to \infty}\alpha \quad x\quad \log(x)^{a(n)-1}/\tilde N(x)\quad =\quad 1\quad. \] The constant \(\alpha>0\) depends only on K and n and \(a(n):=\sum_{1<m| n}\phi(m)\quad(K(\zeta_ m):K)^{-1},\) where \(\zeta_ m\) is a primitive m-th root of unity. For \(K={\mathbb{Q}}\) similar asymptotic estimates have been found by A. M. Baily [J. Reine Angew. Math. 315, 190-210 (1980; Zbl 0421.12007)] and H. Cohn [Proc. Am. Math. Soc. 5, 476-477 (1954; Zbl 0055.269)] for \(n=2,3,4\). For non-Galois cubic extensions see H. Davenport and H. Heilbronn [Proc. R. Soc. Lond., Ser. A 322, 405-420 (1971; Zbl 0212.081)] and for Galois extensions with certain Frobenius groups see H.-D. Steckel [J. Reine Angew. Math. 343, 36-63 (1983; Zbl 0509.12015)].
The second and main result of the paper under review is to describe the distribution of the Frobenius automorphism (\({\mathfrak q},L/K)\) of a fixed prime ideal \({\mathfrak q}\) of K in those cyclic L/K of degree n where \({\mathfrak q}\) is non-ramified. For \(m| n\) let \(N_{{\mathfrak q},m}(x)\) denote the number of these cyclic extensions L/K where in addition \(f_{L/K}\leq x\) and (\({\mathfrak q},L/K)^ m=1\) holds. Then for \(m,m'| n\) one gets \[ \lim_{x\to \infty}N_{{\mathfrak q},m}(x)/N_{{\mathfrak q},m'}(x)\quad =\quad m/m'\quad. \]
Reviewer: C.-G.Schmidt

MSC:
11R45 Density theorems
11N45 Asymptotic results on counting functions for algebraic and topological structures
11R18 Cyclotomic extensions
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