Locally constant group schemes Let $k$ be a field of characteristic $p$, then the group scheme $\mu_\ell$, if $p$ does not divide $\ell$, is étale-locally isomorphic to $\mathbb{Z}/\ell\mathbb{Z}$. I have two quick questions which I don't feel very certain. Is $\mu_p$ fppf locally isomorphic to $\mathbb{Z}/p\mathbb{Z} $? Also, is $\alpha_p $ fppf locally constant?

I guess if $\mu_p$ were locally isomorphic to $\mathbb{Z}/p\mathbb{Z}$, there would be a $k$-algebra $A$ such that $\mu_{p,A}\simeq \mathbb{Z}/p\mathbb{Z}_A$. But for a field $K$ which is a $A$-algebra, $\mu_{p,A}(K)=1$ but $\mathbb{Z}/p\mathbb{Z}_A(K)$ is the abstract group $\mathbb{Z}/p\mathbb{Z}$
