I have read that the goal of Class Field Theory is to characterize all the abelian extensions of a number field $K$ in terms of parameters of $K$.
Let $L|K$ be an abelian extension of number fields. I want to know what are the main theorems of class field theory and what are the corresponding versions of these theroems for the particular case $K=\mathbb{Q}$.
$\newcommand{\Q}{\Bbb Q} \newcommand{\N}{\Bbb N} \newcommand{\R}{\Bbb R} \newcommand{\Z}{\Bbb Z} \newcommand{\C}{\Bbb C} \newcommand{\A}{\Bbb A} \newcommand{\ab}{\mathrm{ab}} \newcommand{\Gal}{\mathrm{Gal}} \newcommand{\prolim}{\varprojlim} $ The good news is that for $K=\Bbb Q$, Kronecker's Jugendtraum (a.k.a. 12th Hilbert problem) is not a dream anymore, thanks to Kronecker–Weber–Hilbert theorem! Namely, a number field $L$ is an abelian extension of $\Q$ if and only if it is contained in some cyclotomic field. From this, one concludes that we have an isomorphism of topological groups $\mathrm{Gal}(\Q^{\ab} / \Q) \cong \widehat{\Bbb Z}^{\times}$, where $\sigma$ is sent to the sequence $(a_n)_{n \geq 1} \in \prolim\limits_{n} (\Z /n \Z)^{\times}$ such that $\sigma(e^{2 i \pi /n}) = e^{2 i \pi a_n/n}$.
How does it relate to the usual statements of global class field theory?
From class field theory, we have a surjective continuous morphism $r : \A_{\Q}^{\times} / \Q^{\times} \to \mathrm{Gal}(\Q^{\ab} / \Q)$ called Artin map. It maps the connected set $\R_{>0} / \Q^{\times}$ to $\mathrm{id}_{\Q^{\ab}}$, since the Galois group is totally disconnected. Then $r$ factors through $$(\A_{\Q, f}^{\times} \times \R^{\times}) / (\Q^{\times} \R_{>0}) \cong (\A_{\Q, f}^{\times} \times \{ \pm 1\}) / \Q^{\times} \cong \A_{\Q, f}^{\times} / \Q^{\times} $$ (here $\A_{\Q, f}$ is the ring of finite adeles, and the second isomorphism comes from the fact that $(x,1)$ is $\Q^{\times}$-equivalent to $(-x,-1)$). One can show that $\A_{\Q, f}^{\times} / \Q^{\times} \cong \widehat{\Z}^{\times}$ (more generally, $K^{\times} \backslash \A_{K, f}^{\times} / \widehat{O_K}^{\times}$ is the class group of $K$). This indicates that $r$ induces actually an isomorphism $\widehat{\Z}^{\times} \cong \Gal(\Q^{\ab} / \Q)$, which is the one we saw above (or its inverse, depending on your definition of $r$).
Another important fact is the existence of the ray class fields. When $m$ is a natural number, we can construct a certain closed subgroup $C_{\Q}^m$ of finite index in $\A_{\Q}^{\times} / \Q^{\times}$ (I should use $(m) \infty$, but since I'm not giving the precise definition, I don't want to introduce unecessary notations — see Neukirch, Class field theory, definition IV.7.2, p. 98). Class field theory associates a finite abelian extension $K_m$ of $\Q$, whose norm subgroup $N_{K_m / \Q}(\A_{K_m}^{\times} / K_m^{\times})$ is $C_{\Q}^m$. One proves (ibid., theorem IV.7.7) that $K_m = \Q(\zeta_m)$. So we get back Kronecker–Weber theorem, since any finite abelian extension is contained in some ray class field!
Some other consequences of Artin reciprocity law (or also Hilbert's "reciprocity" law for Hilbert's symbol) applied to quadratic extenions of $\Q$ is simply Gauss' quadratic reciprocity law!
Moreover, one usually sees Cebotarev theorem included in CFT textbooks. In that case, applying this result to the cyclotomic extension $\Q(\zeta_m) / \Q$, one gets Dirichlet progression theorem!
However, notice that some theorems from class field theory are not very interesting for $K = \Q$. Typically, since $\Z$ is a PID, the class number of $\Q$ is $1$ so the Hilbert class field of $\Q$ is... $\Q$. It tells you that $\Q$ has no non-trivial abelian everywhere unramified extensions. But you already knew that the discriminant of $\Q$ is $1$, so it has no non-trivial unramified extension! Similarly, the principal ideal theorem is completely trivial when $K = \Q$.
