How to write -1 as a square in a finite field of characteristic 2

Question

If $\mathbb{F}_{q}$ is a finite field of characteristic $2$, where $q$ is a power of $2$ and $\beta$ is a generator of $\mathbb{F}^{*}$, then I know that $-1$ is a square in $\mathbb{F}$, but how do I find $1\leq k \leq q-2$ such that $\beta^{k}=-1$?

Many thanks!

Answer 1

If the characteristic is $2$ then $1=-1$. Being $\beta$ a generator of $\mathbb{F}^*$ the smallest $k$ such that $\beta ^k=-1=1$ is $q-1$. Then there's no such $k$ in $\{1,...,q-2 \}$.