Is there an example of a group $G$ which cannot be a Galois group over $\mathbb{Q}$?

Question

I was reading Dummit and Foote and they talked about the study of Class Field Theory, which studies abelian extensions of an arbitrary finite extension of $\mathbb{Q}$ (and related, a theorem that says any abelian $G$ may be realized as the Galois group of some subfield of a cyclotomic extension). So my question is, is there any group that cannot be realized as a Galois group over $\mathbb{Q}$? I know that from above $G$ must be nonabelian, and that if we don't require $\mathbb{Q}$ as our base field specifically, using the Fundamental Theorem of Galois Theory the answer is "no" via realizing $G < S_n$ for some $n$ and then taking the extension $K/K^G$, where $K = \mathbb{Q}[x_1, ... , x_n]$. But if we require the base field to be $\mathbb{Q}$?

Answer 1

Following the remark by Franz Lemmermeyer, we can easily name an infinite group which is not the Galois group of any (necessarily infinite degree) Galois extension. This is the group of integers, $\Bbb Z$. By infinite Galois theory, the Galois group of an infinite-degree field extension $L\mid K$ becomes an infinite compact Hausdorff topological group, which is uncountable. So it cannot be $\Bbb Z$. This holds in particular for $K=\Bbb Q$.