Counting cyclic quartic extensions of a number field (Q819857)

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 \textit{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 \(4\).