This is exercise 2.6 in Milne's "Algebraic Groups". It asks to show that the commutative algebraic group extensions over a field $k$ of characteristic $p$
$$0\to\mu_p\to G\to \mathbb{Z}/p\mathbb{Z}\to 0$$
are classified by the group $k^×/k^{×p}$. However, I am stuck at deciphering what $\mathbb{Z}/p\mathbb{Z}$ is in this context? My first guess was that it's the constant algebraic group corresponding to the abstract group $\mathbb{Z}/p\mathbb{Z}$, but this doesn't seem to give the right answer.
The extensions should be classified by $H^1_{flat}(\mathbb{Z}/p\mathbb{Z},\mu_p)$, which should be calculated by the complex
$Hom_k(k[T]/(T^p-1),\prod_{i\in\mathbb{Z}/p\mathbb{Z}}k)\to Hom_k(k[T]/(T^p-1),\prod_{i,j\in\mathbb{Z}/p\mathbb{Z}}k)\to Hom_k(k[T]/(T^p-1),\prod_{i,j,l\in\mathbb{Z}/p\mathbb{Z}}k),$ which doesn't give the 'correct' answer. Did I guess wrong, and $\mathbb{Z}/p\mathbb{Z}$ means something else in this context?