I will start by saying that I know very little about Class Field Theory, so I am hoping for someone to shed some light on this.
If $K$ is a finite abelian extension of $\mathbb{Q}$, then one can build a map $\phi$ from the set of all but finitely many prime numbers to $\mathbb{C}^\times$ by composing the Artin map with a nontrivial character of the Galois group $\text{Gal}(K/\mathbb{Q})$, i.e. $$ \phi : p \mapsto \left(\frac{K/\mathbb{Q}}{p}\right) \mapsto \chi\left(\left(\frac{K/\mathbb{Q}}{p}\right)\right), $$ and then extend it to almost all integers by multiplicativity.
If the extension has degree 2, then $\phi$ is simply the Kronecker symbol $$ \phi(n) = \left(\frac{\Delta}{n}\right),$$ and by Quadratic Reciprocity, this is a character modulo $\Delta$.
I was wondering if something similar can be done for higher-degree extensions, say if $[K:\mathbb{Q}] = 3$, can we express $\phi$ as a character of some finite group?