Re: Ray Class Fields

[Jay <jayp@MIT.EDU>]

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

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