Remark 39.8.5. Let $k$ be a field of characteristic $p > 0$. Let $\alpha \in k$ be an element which is not a $p$th power. The closed subgroup scheme

\[ G = V(x^ p + \alpha y^ p) \subset \mathbf{G}_{a, k}^2 \]

is reduced and irreducible but not smooth (not even normal).

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