On the fields of rational fractions over $\mathbb{F}_{p}$

Question

Let $K$ be a field with characteristic $p>0$. Let $K^{p}=\left\{ a^{p}:a\in K\right\}$. Prove that $K^p$ is a subfield of $K$. Furthermore, if $K=\mathbb{F}_{p}\left(X\right)$ is the fields of rational fractions over $\mathbb{F}_{p}$, find $K^p$ and $[K:K^p]$.

Help me prove the second part.

Thanks a lot.