A mean value theorem for discriminants of abelian extensions of a number field

Let k be an algebraic number field and let N(k,Cℓ;m) denote the number of abelian extensions K of k with G(K/k)≅Cℓ, the cyclic group of prime order ℓ, and the relative discriminant D(K/k) of norm equal to m. In this paper, we derive an asymptotic formula for ∑m⩽XN(k,Cℓ;m) using the class field theory and a method, developed by Wright. We show that our result is identical to a result of Cohen, Diaz y Diaz and Olivier, obtained by methods of classical algebraic number theory, although our methods allow for a more elegant treatment and reduce a global calculation to a series of local calculations.