Multiplicative Selfinverse in Fields 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?

The equation $x^2-1$ is degree $2$ and thus can have at most two solutions in any field. So checking that $1$ and $-1$ satisfy this is enough to know that they are the only self-inverse elements. (As Nate points out, in the field of characteristic $2$ they are also equal to each other, so there is only one self-inverse element in this case).
