Suppose that $G$ is an algebraic group over a field $k$. Let $G^o$be the connected component of the identity. Since $G^0$ contains a $k$-rational point (the identity) therefore it is geometrically connected. I am wondering whether the remaining connected components of $G$ are also geometrically connected.



Geometrically connected components of an algebraic group
No, take for example $k=\mathbb{Q}$ and $G=\mu\_p$ for some prime $p>2$. Then $G$ has only two connected components, whereas $G\_\overline{\mathbb{Q}}$ has $p$.


