Show that a map is not an automorphism in an infinite field How should I show that a map $f(x) = x^{-1}$ for $x \neq 0$ and $f(0) = 0$ is not an automorphism for an infinite field?
Thanks for any hints.
Kuba

An elemantary way;
Assume that $\phi$ is an automorphism of $F$ as you defined.Notice that if $\phi(x)=x$ then $x=1$ or $x=-1$ or $x=0$.
Now let $r$ be any nonzero elements of $F$ then set $x=r+\dfrac 1r$.
So we have, $\phi(x)=\phi(r+\dfrac 1r)=\phi(r)+\phi(\dfrac 1r)=\dfrac 1r+r=x$ which means that
$r$ must be a root of $$r+\dfrac 1r=0$$
$$r+\dfrac 1r=1$$
$$r+\dfrac 1r=-1$$
You have finite possible $r$ contradicting the fact that $F$ is infinite.
