Let $p$ be prime and denote the field of fractions of $\mathbb{Z}_{p}[x]$ by $\mathbb{Z}_{p}(x)$. Prove that $\mathbb{Z}_{p}(x)$ is an infinite field of characteristic $p$.
I am not sure if there is a simple way of doing this proof. How do I show that it is infinite and a field? I think the characteristic part is easy once it is shown it is an infinite field.