Let $K$ be an algebraic number field, assumed to be Galois, with Galois group
$G = Gal(K/\mathbb{Q})$.
Is knowing the abelianization of $G$ alone, without other information on $K$, enough to determine the ideal class group of $K$? Or can we have two different Galois ANF $K_1$, $K_2$, having the same abelianization of their Galois groups, but non-isomorphic ideal class group?
I have just started learning the subject, so forgive the naiveness of my question.
Edit 1: this was answered below. I wonder if the answer would be any different if instead of
$G = Gal(K/\mathbb{Q})$,
one replaces it with
$H = Gal(\bar{\mathbb{Q}} / K)$,
where $\bar{\mathbb{Q}}$ is the algebraic closure of $\mathbb{Q}$.
No, take e.g. $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-5})$. Both have Galois group $\mathbb{Z}/2\mathbb{Z}$ but $\mathbb{Z}[i]$ is a PID whereas $(2,1+\sqrt{-5})$ is not principal in $\mathbb{Z}[\sqrt{-5}]$.