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单词 ExamplesOfPrimeIdealDecompositionInNumberFields
释义

examples of prime ideal decomposition in number fields


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

Example 1

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

𝒪K=[1+-72]

ThediscriminantPlanetmathPlanetmathPlanetmath of this field is DK/=-7. We look at thedecomposition in prime idealsMathworldPlanetmathPlanetmath of some prime ideals in :

  1. 1.

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

    (7)𝒪K=(-7)2

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

    D((-7)/(7))=Gal(K/)

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

    T((-7)/(7))=Gal(K/)

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

  2. 2.

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

    D(5𝒪K/(5))=Gal(K/)=D(13𝒪K/(13))

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

    T(5𝒪K/(5))={Id}=T(13𝒪K/(13))
  3. 3.

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

    2𝒪K=(2,1+-72)(2,1--72)=𝒫𝒫
    29𝒪K=(29,14+-7)(29,14--7)=

    so e=f=1,g=2 and

    D(𝒫/(2))=T(𝒫/(2))={Id}=D(/(29))=T(/(29))

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

Gal(L/)(/7)×/6

Moreover

Gal(L/)={σa:LLσa(ζ7)=ζ7a,a(/7)×}

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/ isDL/=-75. Let 𝒪L denote the ring ofintegers 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:

\\xymatrixL=(ζ7)\\ar@-[d]3&(1-ζ7)6\\ar@-[d]&𝔓𝔓\\ar@-[d]&(5)\\ar@-[d]&𝔔1𝔔2𝔔3\\ar@-[d]K=(-7)\\ar@-[d]2&(-7)2\\ar@-[d]&(2,1+-72)(2,1--72)\\ar@-[d]&(5)\\ar@-[d]&(13)\\ar@-[d]&(7)&(2)&(5)&(13)

  1. 1.

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

    7𝒪L=(1-ζ7)6=𝔗6

    Thus

    e(𝔗/(7))=6,f(𝔗/(7))=g(𝔗/(7))=1

    Note that, by the properties of the fixed fields of decompositionand inertia groups, we must haveLT(𝔗/(7))==LD(𝔗/(7)), thus, byGalois theory,

    D(𝔗/(7))=T(𝔗/(7))=Gal(L/)
  2. 2.

    The ideal 2 factors in K as above,2𝒪K=𝒫𝒫, and each of theprime ideals 𝒫,𝒫 remains inert from K toL, i.e. 𝒫𝒪L=𝔓, a prime idealof 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):

    e(𝔓/(2))=1,f(𝔓/(2))=3,g(𝔓/(2))=2

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

    D(𝔓/(2))=<σ2>/3,T(𝔓/(2))={Id}
  3. 3.

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

    e(𝔖/(5))=1,f(𝔖/(5))=6,g(𝔖/(5))=1
    D(𝔖/(5))=Gal(L/),T(𝔖/(5))={Id}
  4. 4.

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

    e(𝔔i/(13))=1,f(𝔔i/(13))=2,g(𝔔i/(13))=3
    D(𝔔i/(13))=<σ6>/2,T(𝔔i/(13))={Id}
  5. 5.

    The prime ideal 29 is splits completely in L,

    29𝒪L=123123

    Also 291mod 7, so f=1,

    e(i/(29))=1,f(i/(29)=1,g(i/(29))=6
    D(i/(29))=T(i/(29))={Id}
Titleexamples of prime ideal decomposition in number fields
Canonical nameExamplesOfPrimeIdealDecompositionInNumberFields
Date of creation2013-03-22 13:53:05
Last modified on2013-03-22 13:53:05
Owneralozano (2414)
Last modified byalozano (2414)
Numerical id12
Authoralozano (2414)
Entry typeExample
Classificationmsc 11S15
Related topicDecompositionGroup
Related topicDiscriminant
Related topicNumberField
Related topicPrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ
Related topicPrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ
Related topicExamplesOfRamificationOfArchimedeanPlaces
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