Re: idealprimedec() behavior

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

* Iwao Kimura [2018-05-09 05:50]:
> Hi pari dev,
> 
> I found the following idealprimdec() behavior:
> (12:28) gp > nf=bnfinit(x^2+5);
> (12:28) gp > id=idealprimedec(nf,3)[1];id
> %5 = [3, [-1, 1]~, 1, 1, [1, -5; 1, 1]]
> (12:28) gp > idealfactor(nf,id)
> %6 =
> [[3, [-1, 1]~, 1, 1, [1, -5; 1, 1]] 1]
> 
> (12:29) gp > idtwo=idealtwoelt(nf,id);idtwo
> %7 = [3, [-1, 1]~]
> (12:29) gp > idealfactor(nf,idtwo)
> %8 =
> [ [3, [-1, 1]~, 1, 1, [1, -5; 1, 1]] 1]
> 
> [[3, [1, 1]~, 1, 1, [-1, -5; 1, -1]] 1]
> 
> What I expected is that %6 and %8 are the same.
> Am I missing something?

Short answer: see ??ideal; a two-elements form is not an ideal, you must
replace idtwo by idealhnf(nf, 3, [-1,1]~) for it to be handled as you
expected.

Long answer: a 2-vector component such as idtwo is understood as an
"extended ideal" [I,t], where I is an actual ideal (3 in your case)
and t is an algebraic number corresponding to a principal ideal. The
documentation states that the "principal part" t is suitably taken
into account by a few multiplicative functions, and discarded by all
others. In your case, it is discarded and what you are looking at in %8
is idealfactor(nf, 3).

Cheers,

    K.B.

P.S. The documentation of ??ideal was confusing. I just clarified it in master.
The version at
  http://pari.math.u-bordeaux.fr/dochtml/html/cont_General_number_fields.html
is up to date.

--
Karim Belabas, IMB (UMR 5251)  Tel: (+33) (0)5 40 00 26 17
Universite de Bordeaux         Fax: (+33) (0)5 40 00 21 23
351, cours de la Liberation    http://www.math.u-bordeaux.fr/~kbelabas/
F-33405 Talence (France)       http://pari.math.u-bordeaux.fr/  [PARI/GP]
`