On Computations in Kummer Extensions

Let k be an algebraic number field containing a primitive mth root of unity. An extension K = k([m ] μ) of k with μ k is called a Kummer extension. These extensions have been studied extensively in the past and they play an important role in class field theory. Recently many new algorithms dealing with Kummer extensions emerged. In this paper we will give algorithms to solve two problems, which are of particular interest; the computation of the relative discriminant dK / k and the computation of Hilbert norm symbols.