inertial degree

Let ι:AB be a ring homomorphismMathworldPlanetmath. Let 𝔓B be a prime idealMathworldPlanetmathPlanetmath, with 𝔭:=ι-1(𝔓)A. The algebraPlanetmathPlanetmath map ι induces an A/𝔭 module structureMathworldPlanetmath on the ring B/𝔓. If the dimension of B/𝔓 as an A/𝔭 module exists, then it is called the inertial degree of 𝔓 over A.

A particular case of special importance in number theoryMathworldPlanetmath is when L/K is a field extension and ι:𝒪K𝒪L is the inclusion mapMathworldPlanetmath of the ring of integersMathworldPlanetmath. In this case, the domain 𝒪K/𝔭 is a field, so dim𝒪K/𝔭𝒪L/𝔓 is guaranteed to exist, and the inertial degree of 𝔓 over 𝒪K is denoted f(𝔓/𝔭). We have the formulaMathworldPlanetmathPlanetmath


where e(𝔓/𝔭) is the ramification index of 𝔓 over 𝔭 and the sum is taken over all prime ideals 𝔓 of 𝒪L dividing 𝔭𝒪L. The prime 𝔓 (and also the prime 𝔭) is said to be inert if f(𝔓/𝔭)=[L:K].


Let ι:[i] be the inclusion of the integers into the Gaussian integersMathworldPlanetmath. A prime p in may or may not factor in [i]; if it does factor, then it must factor as p=(x+yi)(x-yi) for some integers x,y. Thus a prime p factors into two primes if it equals x2+y2, and remains prime in [i] otherwise. There are then three categoriesMathworldPlanetmath of primes in [i]:

  1. 1.

    The prime 2 factors as (1+i)(1-i), and the principal idealsMathworldPlanetmath generated by (1+i) and (1-i) are equal in [i], so the ramification index of (1+i) over is two. The ring Z[i]/(1+i) is isomorphic to /2, so the inertial degree f((1+i)/(2)) is one.

  2. 2.

    For primes p1mod4, the prime p factors into the productPlanetmathPlanetmathPlanetmath of the two primes (x+yi)(x-yi), with ramification index and inertial degree one.

  3. 3.

    For primes p3(mod4), the prime p remains prime in [i] and [i]/(p) is a two dimensional field extension of /p, so the inertial degree is two and the ramification index is one.

In all cases, the sum of the products of the inertial degree and ramification index is equal to 2, which is the dimension of the corresponding extensionPlanetmathPlanetmath (i)/ of number fieldsMathworldPlanetmath.

1 Local interpretations & generalizations

For any extension ι:AB of Dedekind domainsMathworldPlanetmath, the inertial degree of the prime 𝔓B over the prime 𝔭:=ι-1(𝔓)A is equal to the inertial degree of 𝔓B𝔓 over 𝔭A𝔭 in the localizationsMathworldPlanetmath at 𝔓 and 𝔭. Moreover, the same is true even if we pass to completions of the local ringsMathworldPlanetmath B𝔓 and A𝔭 at 𝔓 and 𝔭. The preservation of inertial degree and ramification indices with respect to localization is one of the reasons why the technique of localization is a useful tool in the study of such domains.

As in the case of ramification indices, it is possible to define the notion of inertial degree in the more general setting of locally ringed spaces. However, the generalizationsPlanetmathPlanetmath of inertial degree are not as widely used because in algebraic geometryMathworldPlanetmathPlanetmath one usually works with a fixed base field, which makes all the residue fieldsMathworldPlanetmath at the points equal to the same field.

Title inertial degree
Canonical name InertialDegree
Date of creation 2013-03-22 12:38:17
Last modified on 2013-03-22 12:38:17
Owner djao (24)
Last modified by djao (24)
Numerical id 6
Author djao (24)
Entry type Definition
Classification msc 12F99
Classification msc 13B02
Classification msc 11S15
Synonym residue degree
Related topic Ramify
Related topic DecompositionGroup
Defines inert