The classical statement of class field theory The classical statement of class field theory is that for finite Galois extension $L/Q$ the following are equivalent:
a. $L$ is a class field.
b. $L/Q$ is abelian.
c. $L \subset Q[\zeta _n]$ for some $n$.
Here class field means for any unramified prime $p \in Q$ how the prime splits in $L$ depends only on the congruence class of $p$ modulo some modulus $N$ -which is a natural number.
I want to see this statement proven using the modern version either the idelic or the ideal theoretic, as something like a corollary.
$b \iff c$ is Kronecker Weber so that is fine (and the opposite direction is very easy).
It seems that (https://ayoucis.wordpress.com/2015/01/26/a-class-field-theoretic-phenomenon/) proves that $a\iff b$. But I don't quite get the proof. Can someone explain how this works? maybe explain the central idea of the proof?
 A: I think the proof in the link(apart from a few typos) is perfectly detailed and clear. But I try to explain the main idea.
By Hilbert theory, you can understand the decomposition of $p$ in $K$ by looking at the $Gal(K/\mathbb{Q})$ for example $p$ is unramified if and only if the inertia group is trivial. In this case decomposition group is generated by the $Frob_p$ and hence $r=[Gal(K/\mathbb{Q}):(frob_p)]$, and you can also compute f.
so the real question is to understand $Frob_p$. the class field theory says that if $K/Q$ is abelian then under the isomorphism $Gal(K/\mathbb{Q})\to I_\mathbb{Q}/\mathbb{Q}NI_K$, $Frob_p$ goes to the idele $(1,1,...,p,1,1,...)\in I_\mathbb{Q}$. so you have to understand the group generated by $(1,1,...,p,1,1,...)$ which seems even harder! but Kronecker-weber  says that $K\subset \mathbb Q(\zeta_N)$ so by compatibility of reciprocity map, and the explicit description of reciprocity map for the cyclotomic field, we see that we have to understand the group generated by $p$ inside $(\mathbb{Z}/N\mathbb{Z})^*/H$ where $H$ is the subgroup of $Gal(\mathbb{Q}(\zeta_N)/Q)$ which fixes $K$. this shows that $b\to a$.
for the implication $a\to b$ the idea is that by the Chebitarov density theorem you have the density of the primes that spilt completely in $K$. now if every prime that decomposes in $K$ also decomposes in $L$ then $K\subset L$ otherwise the density of the splitting primes in $KL$ would be greater than it should be. now if $K$ is a class field,$p$ splits if and only if $p\in S$ where $S\subset \mathbb{Z}/N\mathbb{Z}$ now $S$ should contain $1$(look at the last paragraph of the proof in the link) and hence every prime that spilt in $\mathbb{Q}(\zeta_n)$ and hence $K\subset \mathbb{Q}(\zeta_n)$ which shows that $a\to c$
(I was a little careless and probably at some places implicitly I assume that $K/\mathbb{Q}$ is Galios. but it is easy to fill the detail by the standard technics of Hilbert theory)
