ray class field
Proposition 1.
Let be a finite abelian extension of number fields, and let be the ring of integers of . There exists an integral ideal , divisible by precisely the prime ideals of that ramify in , such that
where is the Artin map.
Definition 1.
The conductor of a finite abelian extension is the largest ideal satisfying the above properties.
Note that there is a “largest ideal” with this condition because if proposition 1 is true for then it is also true for .
Definition 2.
Let be an integral ideal of . A ray class field of (modulo ) is a finite abelian extension with the property that for any other finite abelian extension with conductor ,
Note: It can be proved that there is a unique ray class field with a given conductor. In words, the ray class field is the biggest abelian extension of with a given conductor (although the conductor of does not necessarily equal !, see example ).
Remark: Let be a prime of unramified in , and let be a prime above . Then if and only if the extension of residue fields is of degree 1
if and only if splits completely in . Thus we obtain a characterization of the ray class field of conductor as the abelian extension of such that a prime of splits completely if and only if it is of the form
Examples:
-
1.
The ray class field of of conductor is the -cyclotomic extension of . More concretely, let be a primitive root of unity. Then
-
2.
so the conductor of is .
-
3.
, the ray class field of conductor , is the maximal abelian extension of which is unramified everywhere. It is, in fact, the Hilbert class field of .
References
- 1 Artin/Tate, Class Field Theory. W.A.Benjamin Inc., New York.
Title | ray class field |
Canonical name | RayClassField |
Date of creation | 2013-03-22 13:54:01 |
Last modified on | 2013-03-22 13:54:01 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 5 |
Author | alozano (2414) |
Entry type | Definition |
Classification | msc 11R37 |
Synonym | conductor |
Related topic | ArtinMap |
Related topic | ExistenceOfHilbertClassField |
Related topic | NumberField |
Related topic | AnExactSequenceForRayClassGroups |
Defines | conductor of an extension |