I assume there are only two multiplicative self inverse in each field with characteristice bigger than $2$ (the field is finite but I think it holds in general). In a field $F$ with $\operatorname{char}(F)>2$ a multiplicative self inverse $a \in F$ is an element such that

$$ a \cdot a = 1.$$

I think in each field it is $1$ and $-1$. Any ideas how to proof that?