splitting and ramification in number fields and Galois extensions

Let F/K be an extension of number fieldsMathworldPlanetmath and let π’ͺF and π’ͺK be their respective rings of integersMathworldPlanetmath. The ring of integers of a number field is a Dedekind domainMathworldPlanetmath, and these enjoy the property that every ideal 𝔄 factors uniquely as a finite product of prime idealsMathworldPlanetmathPlanetmath (see the entry fractional idealMathworldPlanetmathPlanetmath (http://planetmath.org/FractionalIdeal)). Let 𝔭 be a prime ideal of π’ͺK. Then 𝔭⁒π’ͺF is an ideal of π’ͺF. Let us assume that the prime ideal factorization of 𝔭⁒π’ͺF into primes of π’ͺF is as follows:

𝔭⁒π’ͺF=∏i=1r𝔓iei (1)

We say that the primes 𝔓i lie above 𝔭 and 𝔓i|𝔭 (divides). The exponent ei (commonly denoted as e(𝔓i|𝔭)) is the ramification index of 𝔓i over 𝔭. Notice that for each prime ideal 𝔓i, the quotient ringMathworldPlanetmath π’ͺF/𝔓i is a finite field extension of the finite fieldMathworldPlanetmath π’ͺK/𝔭 (also called the residue fieldMathworldPlanetmath). The degree of this extension is called the inertial degree of 𝔓i over 𝔭 and it is usually denoted by:


Notice that as it is pointed out in the entry β€œinertial degree (http://planetmath.org/InertialDegree)”, the ramification index and the inertial degree are related by the formula:

βˆ‘i=1re(𝔓i|𝔭)f(𝔓i|𝔭)=[F:K] (2)

where r is the number of prime ideals lying above 𝔭 (as in Eq. (1)). See the theorem below for an improvement of Eq. (2) in the case when F/K is Galois.

Definition 1.

Let F,K and Pi,p be as above.

  1. 1.

    If ei>1 for some i, then we say that 𝔓i is ramified over 𝔭 and 𝔭 ramifies in F/K. If ei=1 for all i then we say that 𝔭 is unramified in F/K.

  2. 2.

    If there is a unique prime ideal 𝔓 lying above 𝔭 (so r=1) and f(𝔓|𝔭)=1 then we say that 𝔭 is totally ramified in F/K. In this case e(𝔓|𝔭)=[F:K].

  3. 3.

    On the other hand, if e(𝔓i|𝔭)=f(𝔓i|𝔭)=1 for all i, we say that 𝔭 is totally split (or splits completely) in F/K. Notice that there are exactly r=[F:K] prime ideals of π’ͺF lying above 𝔭.

  4. 4.

    Let p be the characteristicPlanetmathPlanetmath of the residue field π’ͺK/𝔭. If ei=e(𝔓i|𝔭)>1 and ei and p are relatively prime, then we say that 𝔓i is tamely ramified. If p|ei then we say that 𝔓i is strongly ramified (or wildly ramified).

When the extension F/K is a Galois extensionMathworldPlanetmath then Eq. (2) is quite more simple:

Theorem 1.

Assume that F/K is a Galois extension of number fields. Then all the ramification indices ei=e(Pi|p) are equal to the same number e, all the inertial degrees fi=f(Pi|p) are equal to the same number f and the ideal p⁒OF factors as:



Title splitting and ramification in number fields and Galois extensions
Canonical name SplittingAndRamificationInNumberFieldsAndGaloisExtensions
Date of creation 2013-03-22 15:05:29
Last modified on 2013-03-22 15:05:29
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 11
Author alozano (2414)
Entry type Definition
Classification msc 12F99
Classification msc 13B02
Classification msc 11S15
Synonym completely split
Synonym strongly ramified
Synonym wild ramification
Related topic Ramify
Related topic InertialDegree
Related topic CalculatingTheSplittingOfPrimes
Related topic PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ
Related topic PrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ
Defines totally ramified
Defines totally split
Defines wildly ramified
Defines tamely ramified