Algebraic Number Field - Ramification

Ramification

Ramification, generally speaking, describes a geometric phenomenon that can occur with finite-to-one maps (that is, maps f: X → Y such that the preimages of all points y in Y consist only of finitely many points): the cardinality of the fibers f−1(y) will generally have the same number of points, but it occurs that, in special points y, this number drops. For example, the map

C → C, z ↦ zn

has n points in each fiber over t, namely the n (complex) roots of t, except in t = 0, where the fiber consists of only one element, z = 0. One says that the map is "ramified" in zero. This is an example of a branched covering of Riemann surfaces. This intuition also serves to define ramification in algebraic number theory. Given a (necessarily finite) extension of number fields F / E, a prime ideal p of O_E generates the ideal pO_F of O_F. This ideal may or may not be a prime ideal, but, according to the Lasker–Noether theorem (see above), always is given by

pO_F = q₁e₁ q₂e₂ ... q_me_m

with uniquely determined prime ideals q_i of O_F and numbers (called ramification indices) e_i. Whenever one ramification index is bigger than one, the prime p is said to ramify in F.

The connection between this definition and the geometric situation is delivered by the map of spectra of rings Spec O_F → Spec O_E. In fact, unramified morphisms of schemes in algebraic geometry are a direct generalization of unramified extensions of number fields.

Ramification is a purely local property, i.e., depends only on the completions around the primes p and q_i. The inertia group measures the difference between the local Galois groups at some place and the Galois groups of the involved finite residue fields.