examples of prime ideal decomposition in number fields

Here we follow the notation of the entry on the decomposition groupMathworldPlanetmath. See also http://planetmath.org/encyclopedia/PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ.htmlthis entry.

Example 1

Let K=(-7); then Gal(K/)={Id,σ}/2, where σ is the complex conjugation map. Let 𝒪K be the ring of integersMathworldPlanetmath of K. In this case:


The discriminantPlanetmathPlanetmathPlanetmath of this field is DK/=-7. We look at the decomposition in prime idealsMathworldPlanetmathPlanetmath of some prime ideals in :

  1. 1.

    The only prime ideal in that ramifies is (7):


    and we have e=2,f=g=1. Next we compute the decomposition and inertia groups from the definitions. Notice that both Id,σ fix the ideal (-7). Thus:


    For the inertia group, notice that σIdmod(-7). Hence:


    Also note that this is trivial if we use the properties of the fixed field of D((-7)/(7)) and T((-7)/(7)) (see the section on “decomposition of extensionsPlanetmathPlanetmath” in the entry on decomposition group), and the fact that efg=n, where n is the degree of the extension (n=2 in our case).

  2. 2.

    The primes (5),(13) are inert, i.e. they are prime ideals in 𝒪K. Thus e=1=g,f=2. Obviously the conjugationMathworldPlanetmath map σ fixes the ideals (5),(13), so


    On the other hand σ(-7)--7mod(5),(13), so σIdmod(5),(13) and

  3. 3.

    The primes (2),(29) are split:


    so e=f=1,g=2 and


Example 2

Let ζ7=e2πi7, i.e. a 7th-root of unityMathworldPlanetmath, and let L=(ζ7). This is a cyclotomic extension of with Galois groupMathworldPlanetmath




Galois theoryMathworldPlanetmath gives us the subfieldsMathworldPlanetmath of L: \xymatrix@dr@C=1pcL=(ζ7)\ar@-[r]\ar@-[d]&(ζ7+ζ76)\ar@-[d](-7)\ar@-[r]&

The discriminant of the extension L/ is DL/=-75. Let 𝒪L denote the ring of integers of L, thus 𝒪L=[ζ7]. We use the results of http://planetmath.org/encyclopedia/PrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ.htmlthis entry to find the decomposition of the primes 2,5,7,13,29:


  1. 1.

    The prime ideal 7 is totally ramified in L, and the only prime ideal that ramifies:




    Note that, by the properties of the fixed fields of decomposition and inertia groups, we must have LT(𝔗/(7))==LD(𝔗/(7)), thus, by Galois theory,

  2. 2.

    The ideal 2 factors in K as above, 2𝒪K=𝒫𝒫, and each of the prime ideals 𝒫,𝒫 remains inert from K to L, i.e. 𝒫𝒪L=𝔓, a prime ideal of L. Note also that the order of 2mod 7 is 3, and since g is at least 2, 23=6, so e must equal 1 (recall that efg=n):


    Since e=1, LT(𝔓/(2))=L, and [L:LD(𝔓/(2))]=3, so

  3. 3.

    The ideal (5) is inert, 5𝒪L=𝔖 is prime and the order of 5 modulo 7 is 6. Thus:

  4. 4.

    The prime ideal 13 is inert in K but it splits in L, 13𝒪L=𝔔1𝔔2𝔔3, and 136-1mod 7, so the order of 13 is 2:

  5. 5.

    The prime ideal 29 is splits completely in L,


    Also 291mod 7, so f=1,

Title examples of prime ideal decomposition in number fields
Canonical name ExamplesOfPrimeIdealDecompositionInNumberFields
Date of creation 2013-03-22 13:53:05
Last modified on 2013-03-22 13:53:05
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 12
Author alozano (2414)
Entry type Example
Classification msc 11S15
Related topic DecompositionGroup
Related topic Discriminant
Related topic NumberField
Related topic PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ
Related topic PrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ
Related topic ExamplesOfRamificationOfArchimedeanPlaces