Is inversion ever a field automorphism? Is the function sending every element of a field to its multiplicative inverse (and 0 to 0) ever a field automorphism?

For $F_2$ and $F_3$ inversion is an automorphism, since it is the identity morphism. I'll prove that the field characteristic has to be $2$ or $3$ for iversion to be an automorphism.
Suppose that $\phi(x)={1 \over x}$ is an automophism. It means that
$$\frac{1}{x+y}=\frac{1}{x}+\frac{1}{y}$$
$$\frac{1}{xy}=\frac{1}{x}\frac{1}{y}$$
The first equation is an interesting one.
Suppose that the field has characteristic other than $2$, and pick $x \neq 0$.
$$\frac{1}{2x}=\frac{1}{x+x}=\frac{1}{x}+\frac{1}{x}=\frac{2}{x}$$
Multiplying by $x$ we arrive at $4=1$, or, equivalently, $3=0$, so the characteristic has to be $3$.
So, the field characteristic has to be $2$ or $3$. 
