Let $k$ be a field and $A$ a finitely generated $k$-algebra. Show that, if $A$ is a field, then $A$ is a finite extension of $k$.

Question

Can anyone help me with the following question?

Let $k$ be a field and $A$ a finitely generated $k$-algebra. Show that, if $A$ is a field, then $A$ is a finite extension of $k$.

I see that $\phi: k \to A$ is injective since $k$ is a field. So, $\frac{k}{(0)} \cong k \cong \phi(k) \subseteq A$. But I am stuck formalizing that $A$ is indeed a finite extension of $k$.