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.
Best Answer
Hint if you take $y \in \mathbb F_p(x)$ how else can you write
$$\underbrace{y+y+\cdots+y}_{\text{p times}}?$$