ramification

When you pass from the rationals up to a bigger number field, a prime number p stops being prime and breaks into several pieces. Usually it breaks into distinct, simple pieces. But sometimes a piece appears with a repeated power, like a double root collapsing — the prime “ramifies.” Ramification is the phenomenon of primes acquiring multiplicities in an extension, and it happens only at finitely many primes.

Concretely, let K be a number field with ring of integers O_K, and let p be a rational prime. In O_K the ideal (p) factors as P_1^{e_1} * ... * P_g^{e_g} into prime ideals. The exponent e_i is the ramification index of P_i over p. We say p ramifies in K if some e_i exceeds 1; otherwise p is unramified. More generally one speaks of ramification of a prime of a base field in any extension.

The primes that ramify are exactly those dividing the discriminant of K, so ramification is controlled by a single integer invariant. Each prime above p also carries a residue degree f_i = [O_K / P_i : Z / p], and the fundamental identity sum of e_i f_i = [K : Q] (for Galois extensions, all e_i equal and all f_i equal). Ramification governs how primes interact with the geometry and Galois theory of the field.