The cyclotomic field $\mathbb{Q}(\zeta_p)/\mathbb{Q}$ is the most basic example of a field extension in which splitting of primes depends on an obvious congruence condition. Specifically, if $\ell$ is another prime, then the Frobenius of $\ell$ is $\ell \mod p \in (\mathbb{Z}/p)^\times = \mathrm{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q})$. In particular, $\ell$ splits in the field iff its Frobenius is trivial, and this is true iff $\ell \equiv 1 \mod p$. We can then relate other congruences to splitting in subfields of $\mathbb{Q}(\zeta_p)$, etc. The theorems of global class field theory show that this basic concept holds in a very general case, although the general case is much harder to prove. This basic example, does, however, motivate the ideas in class field theory, which have greatly influenced modern number theory and related areas. (As an added note, the fact that the Artin reciprocity law is true for cyclotomic fields is actually a key ingredient in the proof for general abelian extensions!)
Answered by Davidac897
