Let $K/\mathbb Q$ be a number field, if there exists a integer $N$ such that for $S \subset \mathbb Z / N\mathbb Z$ $$ \text{Spl} (K/\mathbb Q)=\{ p \ | \ p \mod N \in S \} $$ then $K/\mathbb Q$ is abelian extension and $K\subset \mathbb Q (\zeta _N)$.
This blog https://ayoucis.wordpress.com/2015/01/26/a-class-field-theoretic-phenomenon/ proves this result using Chebotarev density theorem. Is it possible to prove this result by restricting ourselves only to algebraic methods? I mean if someone treated class field theory only from cohomological perspective, is it possible to arrive at this result?
Let me know if you know any reference which does this.
It seems that it is possible to prove Chebotarev density theorem using the Artin reciprocity map (and I expect this proof to be shorter) but I could not find this proof.
Thank you.