I'm taking a class in CFT and I'm trying to figure out what the theorems say and what they can be used for to get a "feel" for them. More explicitly, say I take $\mathbb{Q}_p$, so we have the local Artin homomorphism:
$\theta:\mathbb{Q}_p^\times \to \textrm{Gal}(\mathbb{Q}_p^{ab}/\mathbb{Q}_p)$.
This map is only an isomorphism if the take the profinite completion of $\mathbb{Q}_p^\times$ on the left, but otherwise we have a bijection between the finite-index open subgroups of $\mathbb{Q}_p^\times$ and abelian extensions of $\mathbb{Q}_p$.
Let's say we take the subgroup $\mathbb{Q}_p^{\times 2}$ of $\mathbb{Q}_p^\times$ and for simplicity let's assume that $p\neq 2$. Can we actually somehow explicitly write down the abelian extension that this subgroup corresponds to? None of the theorems look constructive, but because they are considered so useful, I guess there has to be ways of writing down the corresponding abelian extension explicitly through generators?
I still need to look into profinite groups, but my understanding is that for the previous example the order of the abelian extension corresponding to $\mathbb{Q}_p^{\times 2}$ has order $(\mathbb{Q}_p^\times:\mathbb{Q}_p^{\times 2})$.
If this is possible, does anyone know of a source where I could find examples of this?