$G$ is a finite group. Show that there are finite extensions $F/\mathbb{Q}$ and $E/F$ such that $E/F$ is Galois with Galois group $G$

Question

I'm self studying fields and Galois theory, and I'm trying to solve the following problem:

Let $G$ be a finite group. Show that there are finite extensions $F/\mathbb{Q}$ and $E/F$ such that $E/F$ is Galois with Galois group $G$.

I'm lost on how to show this. How should I get started on this problem?

Thanks in advance, any suggestions would be greatly appreciated.

Answer 1

Going by this answer, for any $n$ there is a finite Galois extension of $\Bbb Q$ with Galois group $S_n$. Let $E$ be such an extension for $S_{|G|}$. Now use the fundamental theorem of Galois theory together with the fact that any group is isomorphic to a subgroup of the full group of permutations of its elements.