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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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