Let $F := \mathbb{F}_p(x)$, the field of rational functions in one variable over the prime field $\mathbb{F}_p$. How can we show that $F$ is an infinite field of finite characteristic?

*Thoughts so far*

$F$ is clearly infinite since (for example) $1,x,x^2,x^3 \ldots$ etc. are all contained in $F$.

Suppose that $f \in F$. Then $f=\frac{p_1(x)}{p_2(x)}$ with $p_1,p_2$ having coefficients in $\mathbb{F}_p$. I'm not sure how we can show that $\text{char}F=p$ though.