I'm wondering when exactly a Galois group is an algebraic group. I know that they are algebraic groups when the extension is finite - so for example, $Gal(\mathbb{C}/\mathbb{R})$ is an algebraic group, but $Gal(\mathbb{R}/\mathbb{Q})$ is not... What are some other examples of algebraic Galois groups? And how can I further classify them and describe their properties?
Thank you :)