Basic Galois theory question

Question

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.

Answer 1

Hint

Let $\psi:Gal(\bar K/\mathbb Q)\longrightarrow Gal(K/\mathbb Q)$ be the restriction, i.e. $\psi(\sigma )=\sigma |_K$. It's easy (more or less... it's in fact important to prove this result) to see that $\psi$ is surjective and that $\ker \psi=Gal(\bar K/K)$, what gives you the isomorphism.