prime splitting
An ordinary prime like 5 is prime in the integers, but once we move into a larger number field it may no longer be prime: it can break apart into several new primes, stay stubbornly whole, or collapse with multiplicity. Understanding which of these happens, for each rational prime and each field, is one of the central computational and theoretical tasks of the subject.
Let K be a number field of degree n with ring of integers O_K, and let p be a rational prime. The ideal (p) factors as P_1^{e_1} * ... * P_g^{e_g} into distinct prime ideals, with ramification indices e_i and residue degrees f_i = [O_K / P_i : F_p], satisfying the fundamental identity sum of e_i f_i = n. Three pure behaviors are named: p splits (completely) when there are n distinct primes each with e = f = 1; p is inert when (p) stays prime (g = 1, f = n); p ramifies when some e_i > 1.
When O_K = Z[a] and m(x) is the minimal polynomial of a, the factorization of (p) mirrors the factorization of m(x) modulo p (the Kummer-Dedekind theorem), for all but the finitely many primes dividing the index. In a Galois extension the splitting is governed by the Frobenius element and, for abelian extensions, by congruence conditions — the seed of class field theory.
In Q(i), an odd prime p splits if p ≡ 1 (mod 4) and is inert if p ≡ 3 (mod 4): for example 5 = (2 + i)(2 - i) splits, while 7 stays prime. This recovers Fermat's theorem on sums of two squares.
The splitting behavior of p in Q(i) depends only on p modulo 4 — the simplest case of a reciprocity law.