|Bill Allombert on Thu, 8 May 2003 22:59:16 +0200|
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|Re: Ray Class Fields|
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,,[~]] > > 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,]) %2 = [36, , [~]] For any prime p>2, bnrclass(Q,[p,]) 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.