$K$ is a field with $2^d$ elements. $K^\*$ is a cyclic group generated by $\alpha$ and $T:K\rightarrow \mathbb F\_2$ is a non-zero $\mathbb F\_2$ linear map.

Can you help me proving that the symmetric $\mathbb F\_2$-bilinear form $K\times K\rightarrow \mathbb F\_2: (x,y)\rightarrow T(xy)$ is non-degenerate, i.e $T(xy)=0$ for all $y\in K$ implies $x=0$

I do not see how this follows following the properties of the bilinear form.