Let $E$ be the splitting field of an irreducible $f(x)\in \mathbb{Q}[x]$, $Gal(E/\mathbb{Q})$ is abelian. Prove that $E = \mathbb{Q}(a)$, where $a$ is one of the roots of $f(x)$.
My attempt:
Let $K$ be any intermediate field between $E$ and $\mathbb{Q}$, then
1) is it true that $K/\mathbb{Q}$ is a Galois extension $\iff$ $Gal(E/K)$ is a normal subgroup of $Gal(E/\mathbb{Q})$?
Because if it is, then $\mathbb{Q}(a)$ being an intermediate field between $E$ and $\mathbb{Q}$ would lead lead to $\mathbb{Q}(a)/\mathbb{Q}$ being a Galois extension (because every subgroup of an Abelian group $Gal(E/\mathbb{Q})$ is normal and thus $Gal(E/\mathbb{Q}(a))$ is a normal subgroup of $Gal(E/\mathbb{Q})$). This means all roots of $f(x)$ are in $\mathbb{Q}(a)\implies E\subset \mathbb{Q}(a)$ and clearly $Q(a)\subset E$. $\therefore E = \mathbb{Q}(a)$.