Question about the abelian extension over $\mathbb{Q}$

Question

Let $f(x)$ be a polynomial over $\mathbb{Q}$, and let $K$ be the splitting field of $f(x)$ over $\mathbb{Q}$.

Is it true that if $K$ is either totally real or CM-field, then the Galois group $G(K/\mathbb{Q})$ is abelian?

If not, what if the hypothesis is changed to "irreducible" polynomial $f(x)$ over $\mathbb{Q}$?

Actually, i'm wonder about when is the Galois group $G(K/\mathbb{Q})$ (of $f(x)\in\mathbb{Q}[x]$ with splitting field $K$ over $\mathbb{Q}$) abelian?

I would be grateful if you could give me any advice or references. Thank you.

Answer 1

The Kronecker-Weber Theorem says that the finite abelian extensions of $\mathbb Q$ (so finite Galois extensions having abelian Galois group) are precisely the subfields of cyclotomic extensions (so those extensions given by adjoining a root of unity).