Jay on Wed, 7 May 2003 19:10:44 -0400 |
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Re: Ray Class Fields |
On Wednesday, May 7, 2003, at 05:49 PM, A. Lozano-Robledo wrote:
?Q=bnfinit(x); ?bnrclass(Q,37) = [18,[18],[[2]~]] What am I doing wrong? Am I defining correctly the field of rational numbers?
Alvaro,You are correctly defining everything above, and PARI is comuting it correctly. The problem is that ray class field for Q with conductor 37 is the _maximal real subfield_, of index 2, of Q(37th roots). You need to tell bnrclass to produce ramification at infinity.
Note that your input is equivalent to gp > Q=bnfinit(x); gp > bnrclass(Q,[37,[0]]) = [18, [18], [[2]~]]as here I have explicitly told PARI not to produce any ramification at the one infinite place, and this was implicitly assumed in your computation. The following input forces ramification:
gp > bnrclass(Q,[37,[1]]) = [36, [36], [[2]~]] and gives the field you desire as well.I should also note that using bnrclass was tremendously faster for producing Q(37th roots) than bnfinit was.
I hope this helps, Jay --- Jonathan Pottharst 333 Western Ave. #2 Cambridge, MA 02139 617-251-8470 email: jayp@mit.edu aim: sharlaon icq: 4374738 http://www.mit.edu/~jayp/ pgp public key at http://www.mit.edu/~jayp/public/pgpkey