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 \(4\).


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


Zbl 0628.12006
Full Text: DOI Numdam EuDML


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