examples of prime ideal decomposition in number fields
Here we follow the notation of the entry on the decomposition group. See also http://planetmath.org/encyclopedia/PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ.htmlthis entry.
Example 1
Let ; then , where is the complex conjugation map. Let be the ring of integers of . In this case:
The discriminant of this field is . We look at the decomposition in prime ideals of some prime ideals in :
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1.
The only prime ideal in that ramifies is :
and we have . Next we compute the decomposition and inertia groups from the definitions. Notice that both fix the ideal . Thus:
For the inertia group, notice that . Hence:
Also note that this is trivial if we use the properties of the fixed field of and (see the section on “decomposition of extensions” in the entry on decomposition group), and the fact that , where is the degree of the extension ( in our case).
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2.
The primes are inert, i.e. they are prime ideals in . Thus . Obviously the conjugation map fixes the ideals , so
On the other hand , so and
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3.
The primes are split:
so and
Example 2
Galois theory gives us the subfields of :
The discriminant of the extension is . Let denote the ring of integers of , thus . We use the results of http://planetmath.org/encyclopedia/PrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ.htmlthis entry to find the decomposition of the primes :
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1.
The prime ideal is totally ramified in , and the only prime ideal that ramifies:
Thus
Note that, by the properties of the fixed fields of decomposition and inertia groups, we must have , thus, by Galois theory,
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2.
The ideal factors in as above, , and each of the prime ideals remains inert from to , i.e. , a prime ideal of . Note also that the order of is , and since is at least , , so must equal (recall that ):
Since , , and , so
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3.
The ideal is inert, is prime and the order of modulo is . Thus:
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4.
The prime ideal is inert in but it splits in , , and , so the order of is :
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5.
The prime ideal is splits completely in ,
Also , so ,
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 |