Algebraic closures and monomorphisms

Question

Let $F$ be a field and $f\in F[X]$, an irreducible polynomial over $F$, $C$ an algebraic closure of $F$ and $a,b \in C$ two roots of $f$.

In an exercise I proved the existence of an $F-$monomorphism $\tau :C \longrightarrow C$ such that $\tau (a)=b$.

But I also have to prove that $\tau$ is an automorphism (from $C$ into $C$).

I am a little bit stuck for ideas to prove this last one and it's ridiculous because I think I proved the hard part...

Can someone help me please?

Thank you in advance!

Answer 1

Hint: Any element $c\in C$ is algebraic over $F$ and hence is a root of some nonzero polynomial $g\in F[X]$. Observe that $\tau$ must map roots of $g$ to roots of $g$, and there are a finite number of roots of $g$ in $C$.

(More generally, this line of argument shows that if $C$ is any algebraic extension of a field $F$ and $\tau:C\to C$ is an $F$-monomorphism, then $\tau$ is surjective.)