I am working at a Problem for some time and it comes down to the question:
Let $K:=\mathbb{Q}_2[\zeta]$ be a cyclotomic extention of the dyadic field $\mathbb{Q}_2$. For any $c \in \mathbb{Q}_2[\zeta+\zeta^{-1}]$ one can define the trace form $b: K\times K \rightarrow \mathbb{Q}_2$, $(x,y) \mapsto trace (c x\overline{y})$. A standard computation shows that $b$ is a bilinear form and $(K,b)$ is a quadratic space. Lets further suppose the dimension of the $\mathbb{Q}_2$-Space $\mathbb{Q}_2[\zeta]$ is even.
My Problem is, that I can not figure out, wether this space is hyperbolic or not. One possible way to do it, is the computation of the Determinant, but I also don't know how to do that. Does anyone have an idea?
Laura Duebel