The Stacks project

Remark 39.7.5. Warning: The result of Lemma 39.7.4 does not mean that every irreducible component of $G/k$ is geometrically irreducible. For example the group scheme $\mu _{3, \mathbf{Q}} = \mathop{\mathrm{Spec}}(\mathbf{Q}[x]/(x^3 - 1))$ over $\mathbf{Q}$ has two irreducible components corresponding to the factorization $x^3 - 1 = (x - 1)(x^2 + x + 1)$. The first factor corresponds to the irreducible component passing through the identity, and the second irreducible component is not geometrically irreducible over $\mathop{\mathrm{Spec}}(\mathbf{Q})$.