Let $F$ be a field and $t$ be a variable. Let $F(t)$ be the field of quotients of the polynomial ring $F[x]$.
Question. What are all the elements in $F(t)$ which are algebraic over $F$?
I think the answer to the above question is just $F$. For if $p(t)/q(t)$ be an element of $F(t)$ which satisfies an irreducible monic polynomial $a(x)\in F[x]$ of degree at least $2$. We may assume that the greatest common divisor of $p$ and $q$ is $1$.
Let us say $a(x)=x^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0$. Substituting $p/q$ in $a(x)$, we see that
$$p^n+a_{n-1}p^{n-1}q+\cdots a_1pq^{n-1}+ a_0q^n=0$$
Since $a_0\neq 0$, we have $p$ divides $q^n$ and $q|p^n$. Therefore $p|q$ and $q|p$ showing that $p=q$ and we are done.
Are there any other ways to see this?