stylistic:style_transfer, structural:expansion
Ramification

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

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 K / L {\displaystyle K/L}, a prime ideal p of O L {\displaystyle {\mathcal {O}}_{L}} generates the ideal pO of O K {\displaystyle {\mathcal {O}}_{K}}. This ideal may or may not be a prime ideal, but, according to the Lasker–Noether theorem (see above), always is given by

with uniquely determined prime ideals q of O K {\displaystyle {\mathcal {O}}_{K}} and numbers (called ramification indices) e. Whenever one ramification index is bigger than one, the prime p is said to ramify in K {\displaystyle K}.

The connection between this definition and the geometric situation is delivered by the map of spectra of rings S p e c O K → S p e c O L {\displaystyle \mathrm {Spec} {\mathcal {O}}_{K}\to \mathrm {Spec} {\mathcal {O}}_{L}}. 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. The inertia group measures the difference between the local Galois groups at some place and the Galois groups of the involved finite residue fields.