Show that $F(\alpha)=F(\beta)$ if $\alpha$ and $\beta$ are roots of the same irreducible polynomial $g$ over field $F$
I defined a map $\psi: F(\alpha) \to F(\beta)$ by sending $p(\alpha)$ to $p(\beta)$. This map is clearly onto. If $p_1(\alpha)=p_2(\alpha)$, then by minimality of the polynomial $g$, we must have that $g$ divides $p_1-p_2$ and hence $p_1(\beta)=p_2(\beta)$. The same argument backward gives that this map is one-one. It is also easy to see that $\psi$ is a homomorphism. Thus, $\psi$ is an isomorphism.
I showed that both these fields are isomorphic. Is this what the question wants me to show? Is there any other way to do it?
Thanks for the help!!