Find the degree of the extensions $\mathbb{F}(x) / \mathbb{F}$

Question

Here $\mathbb{F}$ is a field and $\mathbb{F}(x) =\{ \frac{q}{p}: q,p \in \mathbb{F}[x], p \neq 0\}$.

I think the proper way to solve is to use the fact that an infinite extension of a countable field should be of infinite degree. However, $\mathbb{F}$ is not necessarily countable.