Let $K$ be a field of characteristic $p > 0$ and $L = K(Y)$. For every $a \in K$, we define $\sigma_a \in \operatorname{Aut}_K(L)$ via $\sigma_a(Y) = Y + a$. Let $H = \{\sigma_a\; | \; a \in \mathbb{F}_p\}$. Show that $L^H = K(Z)$ where $L^H$ is the fixed field of $H$ and $Z = Y^p - Y$.
I don't really know where to start with either $L^H \subset K(Z)$ or $K(Z) \subset L^H$. If $a \in K(Z)$, I have to show that $\sigma(a) = a$ for every $\sigma \in H$ but how exactly can I do this?. For the other inclusion, I'm kinda confused what it means, that $a \in K(Z)$. Any hints would be greatly appreciated.