Trace forms of normal extensions over local fields

Let k be a local field of char(k)≠2 and K/k a finite field extension of degree n. Then K can be viewed as a quadratic space of k under the quadratic form T(X) =tr_K/k(x²). The invariants of this form are given in the case when K/k is a Galois extension, except for Galois extensions K/k with k dyadicn divisible by 4 and the 2-Sylowgroups of the Galois group are non-cyclic. Conversely all quadratic forms of a local field k of char(k)≠ 2 which appear as trace forms of Galois extensions of k are determined.