TITLE: If Gal(K,Q) is abelian then |Gal(K,Q)|=n
QUESTION [10 upvotes]: Let $f(x)\in  \mathbb Q[x]$ irreducible of degree $n$ and $K$ its splitting field over $\mathbb Q$. Prove that if $\operatorname{Gal}(K/\mathbb Q)$ is abelian, then $|\operatorname{Gal}(K/\mathbb Q)|=n$.
How can I prove this?

REPLY [21 votes]: I wonder if this works out:
Since $K/ℚ$ is abelian, every intermediate extension is normal and so is $ℚ(α)$ for some zero $α ∈ K$ of $f$. This must mean that $ℚ(α)$ is a splitting field of $f$ and so $K = ℚ(α)$.