Let $k$ a field. Let $\operatorname{char}(k) \not\mid n$.

Consider the two group schemes $\mu\_n=spec(k[t]/(t^n-1))$ and $\underline{\mathbb{Z}/n\mathbb{Z}}$ the constant group scheme associated to the cyclic group ${\mathbb{Z}/n\mathbb{Z}}$. Are they isomorphic as group schemes? Do they become isomorphic when $k$ is algebraically closed? 

It is false in general because $\mu\_n(k)$ may not have $n$ elements. But when are they isomorphic?