Re: Ray Class Fields

[Bill Allombert <allomber@math.u-bordeaux.fr>]

On Wed, May 07, 2003 at 05:49:10PM -0400, A. Lozano-Robledo wrote:
> I am having trouble using the Class Field Theory functions.
> 
> If I type:
> 
> ?P=(x^37-1)/(x-1);
> ?C37=bnfinit(P);
> ? C37.clgp.no
> %1 = 37
> 
> which is correct, since 37 is an irregular prime.
> However, if I build the cyclotomic extension by class field theory:
> 
> ?Q=bnfinit(x);
> ?bnrclass(Q,37)
>  = [18,[18],[[2]~]]
> 
> What am I doing wrong? Am I defining correctly the field of rational
> numbers?

The first command compute Cl(Q(zeta_37)).
The second command compute the ray class group Cl_37(Q).

This is unrelated, even if you do not forget the archimedean place and
compute Cl_37.oo(Q):

?bnrclass(Q,[37,[1]])
%2 = [36, [36], [[2]~]]

For any prime p>2,
bnrclass(Q,[p,[1]]) will output
[p-1, [p-1], ...]
since Cl_p.oo(Q)=(Z/pZ)^*.

This is obviously false for Cl(Q(zeta_p)).

Obviously Cl(Q(zeta_p)) is much harder to compute than Cl_p.oo(Q).

Cheers,
Bill.