examples of prime ideal decomposition in number fields
Here we follow the notation of the entry on the decompositiongroup. 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:
Thediscriminant of this field is . We look at thedecomposition in prime ideals
of some prime ideals in :
- 1.
The only prime ideal in that ramifies is :
and we have . Nextwe compute the decomposition and inertia groups from thedefinitions. Notice that both fix theideal . Thus:
For the inertia group, notice that . Hence:
Also note that this is trivial if we use the properties of thefixed field of and (seethe section on “decomposition of extensions
” in the entry ondecomposition group), and the fact that , where is the degree of the extension ( in our case).
- 2.
The primes are inert, i.e. they are prime idealsin . Thus . Obviously the conjugation
map fixes the ideals , so
On the other hand , so and
- 3.
The primes are split:
so and
Example 2
Let , i.e. a -root of unity,and let . This is a cyclotomic extension of with Galois group
Moreover
Galois theory gives us the subfields
of :
The discriminant of the extension is. Let denote the ring ofintegers of , thus . We use the results of http://planetmath.org/encyclopedia/PrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ.htmlthis entry to find the decomposition of the primes :
- 1.
The prime ideal is totally ramified in , and theonly prime ideal that ramifies:
Thus
Note that, by the properties of the fixed fields of decompositionand inertia groups, we must have, thus, byGalois theory,
- 2.
The ideal factors in as above,, and each of theprime ideals remains inert from to, i.e. , a prime idealof . Note also that the order of is , and since is at least , , so must equal (recall that ):
Since , , and , so
- 3.
The ideal is inert, isprime and the order of modulo is . Thus:
- 4.
The prime ideal is inert in but it splits in,, and , so the order of is :
- 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 |