ray class field
Note that there is a “largest ideal” with this condition because if proposition 1 is true for then it is also true for .
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 ).
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
so the conductor of is .
, 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 .
- 1 Artin/Tate, Class Field Theory. W.A.Benjamin Inc., New York.
|Title||ray class field|
|Date of creation||2013-03-22 13:54:01|
|Last modified on||2013-03-22 13:54:01|
|Last modified by||alozano (2414)|
|Defines||conductor of an extension|