Can anyone help with this:
If $L/K$ is a finite field extension, and we have a $K$-bilinear form given by $(x,y)\mapsto Tr_{L/K}(xy)$ then the form is either non-degenerate or $Tr_{L/K}(x)=0$ for every $x\in L$.
So far, I feel like I've run into something a little nonsensical. Suppose the form is degenerate, i.e. $\exists \alpha\in L$, $\alpha\neq 0$ such that $(\alpha,\beta)=0$ for all $\beta$. Then specifically, $Tr_{L/K}(\alpha\alpha^{-1})=Tr_{L/K}(1)=0$. But it is a theorem from Morandi's "Field and Galois Theory" that if $\alpha\in K$ (the base field) then $Tr_{L/K}(\alpha)=n\alpha$ where $n$ is the dimension of the field extension, so in this case we get that $[L:K]=0$ which doesn't make any sense. Am I doing something wrong here?
Thanks!