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Counting cyclic quartic extensions of a number field. (English) Zbl 1090.11068

Let \(G\) be a finite group and \(K\) some number field. Let \(N_K(G,X)\) denote the number of all Galois extensions \(L/K\) with Galois group \(G\) such that the norm of the relative discriminant is \(\leq X\). By a conjecture of Malle it is expected that there are constants \(a_K, b_K, c_K\) (depending on \(G\)) such that \[ N_K(G,X) = c_K X^{a_K} (\log X)^{b_K-1}. \] The constants \(a_K\) and \(b_K\) were determined by D. J. Wright [Proc. Lond. Math. Soc. (3) 58, No. 1, 17–50 (1989; Zbl 0628.12006)] for all abelian groups; in this article, the authors determine \(c_K\) in the case where \(G\) is the cyclic group of order \(4\).

MSC:

11R16 Cubic and quartic extensions
11R45 Density theorems
11R29 Class numbers, class groups, discriminants
11R32 Galois theory

Citations:

Zbl 0628.12006
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References:

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