Re: problem with nfhilbert

["Karim.Belabas" <Karim.Belabas@math.u-psud.fr>]

On Tue, 17 Apr 2001, Walter Hofmann wrote:
> I'm trying to calculate norm residue symbols in the field K=Q_2(i).
> Since adjorning sqrt(3) to K is an unramified extension, prime elements
> like 1+i are no norms. Thus the local norm residue symbol (1+i,3) is -1.
> 
> I use the following lines in gp (2.1.0) to check this:
> 
> nf=nfinit(x^2+1)
> p=idealprimedec(nf,2)[1]
> nfhilbert(nf,1+x,3,p)
> 
> But this gives "+1"! Is this a bug in pari or is there an error with my
> usage of gp?

I'd call this a bug, which can be corrected with an "advanced" usage of GP.
For relative number field computations, GP expects "elements in nf", where
'nf', as all base fields, is expected to have a variable of LOWEST priority
than polynomials defining relative extensions.

Here,

  nfhilbert(subst(nf,x,y), 1+y, 3, p)   [or start directly with nfinit(y^2+1)]

gives the expected answer.

Of course, this is quite confusing since the relative polynomial is only
implicit there. Also the nfhilbert() code should check its input arguments
anyway, and not give a bogus answer.

All the more confusing since this could only occur at a prime above 2 [the
problematic code handles only this special case]

I've changed a bit the internal logic so that your input is accepted in any
case.

      Karim.
-- 
Karim Belabas                    email: Karim.Belabas@math.u-psud.fr
Dep. de Mathematiques, Bat. 425
Universite Paris-Sud             Tel: (00 33) 1 69 15 57 48
F-91405 Orsay (France)           Fax: (00 33) 1 69 15 60 19
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