On the norm groups of Galois 2n-extensions of algebraic number fields

Let K be a finite extension of a p-adic number field k. By local class field theory there is only a finite number of norm subgroups of the multiplicative group k* of k that contain the norm group Nk/kK*. If X is a subgroup of a group Y, then the interval (X, Y) is the set of subgroups of Y that contain X including X and Y. In the present work we investigate the number of norm groups in the interval (Nk/kK*,k*) for a given finite Galois extension K/k of algebraic number fields. There are finite Galois 2-extensions and Galois extensions of odd degrees such that the corresponding intervals contain only a finite number of norm groups. The main theorem, however, states that for any finite Galois extension K/k of even degree that is not a 2-extension, called 2n-extension, the interval (NK/kK*, k*) contains infinitely many norm groups. (C) 2008 Elsevier Inc. All rights reserved.