I'm trying to solve this question:
If $K$ is a field and $f:K\to K$ is defined by $f(0)=0$ and $f(x)=x^{-1}$ for $x\neq 0$, show $f$ is an automorphism of $K$, if and only if, $K$ has at most four elements.
The converse seems easy, but I'm really stuck in the first implication.
Anyone has an idea?
Thanks