Let $p$ be a prime number and consider the ring $\mathbb{F}_P=\{0,1,\ldots,p-1\}$ of integers modulo $p$, which is a field. It follows that the polynomal ring $\mathbb{F}_p[x]$ is an integral domain, and so we can consider the field of fractions $\text{Frac}\left(\mathbb{F_p}[x]\right)$ of $\mathbb{F}_p[x]$. Now, since $\mathbb{F}_p \subseteq \text{Frac}\left(\mathbb{F_p}[x]\right)$, $\text{Frac}\left(\mathbb{F_p}[x]\right)$ becomes a field extension of $\mathbb{F}_p$.
Problem: Prove that $\big[\text{Frac}\left(\mathbb{F_p}[x]\right):\mathbb{F}_p \big]=\infty$ is true.
I don't know where to start to prove this statement. I have a feeling that a proof by contradiction, utilizing the fact that the polynomial ring as a vector space is of infinite dimension, might be a course to take.
Any finite extension of $\Bbb{F}_p$ is finite. The field of fractions contains $\Bbb{F}_p[x]$ as a subring, so it is infinite.