Let $k$ be an algebraically closed field of characteristic $p$.
What is the normalization of $k[z][t]/(t^p+t^{p-1}z-z)$?
The normalization of the Spec of this ring describes the restriction of the standard Artin-Schreier cover of $\mathbb{P}^1_k$, to the complement of $0$ (here $t$ is a uniformizer at $\infty$). Thus, since the Artin-Schreier cover is genus 0, the normalization should be isomorphic to $k[T]$. Since this ring doesn't seem to be isomorphic to $k[T]$, I don't think it's normal.
Hence, what is its normalization? (And what is the isomorphism from $k[T]$ to its normalization?)