Let X be a subgroup of a group Y. The interval (X, Y) is the set of subgroups of Y that contain X including X and Y. By local class field theory the interval (NK/kK*, k*) contains a finite number of norm groups for any finite extension K of a p-adic number field k. In the present work we investigate the number of norm groups in the interval (NK/kK*, k*) for a given finite extension K I k of algebraic number fields. We prove that if K/k is an extension of prime degree, or of degree tt such that the normal closure of K over k has the Galois group isomorphic to A or S, then the interval (NK/kK*, k*) contains only the obvious two norm groups. Also, the interval (NK/kK*, k*) contains a finite number of norm groups for any Galois extension of degree 4, and there are extensions with Galois groups order 8 for which the corresponding intervals contain a finite number of norm groups. The main theorem in our earlier work states that the interval (NK/kK*, k*) contains infinitely many norm groups for any Galois extension of even degree that is not a 2-extension, the so-called 2n-extensions. In the present work we generalize the main theorem to non-Galois 2n-extensions K/k, and determine some subintervals of (NK/kK*, k*) that contain infinitely many norm groups. We then use this theorem to prove that the interval (NK/kK*, k*) contains infinitely many norm groups for any Galois 2-extension Klk with the Galois group that either contains an element of order 8 or contains the quaternion group Q(8) of order 8, or Q(8) is a homomorphic image of the Galois group.
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