Local trace forms

Let F be a local field of characteristic ≠2 and K a Galois extension field of F of degree n. Then K can be viewed as a quadratic space over F under the bilinear form T(xy)=tr_K/Fxy for xyεK. The invariants of this form are given in the case when n is odd; when n is even and F is nondyadic; and when n is evesF dyadic, and K/F is unramifed.