Let $K/\mathbb{Q}$ be a Galois extension, and $\bar{K}$ the algebraic closure of $K$.
If we consider the following grups:
- $G \cong Gal(\bar{K}/\mathbb{Q})$
- $H \cong Gal(\bar{K}/K)$
Then $H$ is a normal subgroup of $G$, and applying the main theorem of Galois theory,
$$Gal(K/\mathbb{Q}) \cong G/H$$
Is this correct?
Thank you in advance.