Re: Potential inconsistent results with fundamental units and KASH 3

[Georgi Guninski <guninski@guninski.com>]

Thank you! Dumb me...


On Sat, Jul 14, 2012 at 11:03:50AM +0200, Karim Belabas wrote:
> * Georgi Guninski [2012-07-14 09:56]:
> > Probably I am missing something, but get strange results.
> > 
> > In 2.5.1 via sage:
> > 
> > pr.<x>=ZZ[];K.<a>=NumberField(x^3 - 6*x^2 + 9*x + 1);K.unit_group().gens()
> > [-1, a^2 - 3*a]
> > 
> > in KASH 3:
> > 
> > f:=X^3 - 6*X^2 + 9*X + 1;nf:=NumberField(f);nu:=UnitGroup(nf);gn:=Generators(nu);  Apply(x->nu.ext1(x),List(Generators(nu))); 
> > 
> > [ [3, -1, 0], [-1, 0, 0] ] # 3 - a, -1
> > 
> > $-1$ is torsion in both cases.
> > 
> > Is this result consistent?
> 
> Yes.
> 
> > If g1= a^2 - 3*a and g2=3 - a I naiively expect to be able to solve either
> > 
> > (+/- g1)^n = +/- g2
> > or
> > (+/- g2)^n = +/- g1
> 
> With n in Z, yes.
> 
> > Can't solve either for n<=10^4 and the coefficient are growing quite fast.
> 
> Try n = -1.
> 
> Btw, bnfisunit() solves this kind of question for you.
> 
> (10:46) gp > K=bnfinit(x^3 - 6*x^2 + 9*x + 1);
> (10:46) gp > K.fu
> %2 = [Mod(x^2 - 3*x, x^3 - 6*x^2 + 9*x + 1)]
> (10:46) gp > bnfisunit(K,3-x)
> %3 = [-1, Mod(0, 2)]~
> 
> Cheers,
> 
>     K.B.
> -- 
> Karim Belabas, IMB (UMR 5251)  Tel: (+33) (0)5 40 00 26 17
> Universite Bordeaux 1          Fax: (+33) (0)5 40 00 69 50
> 351, cours de la Liberation    http://www.math.u-bordeaux1.fr/~belabas/
> F-33405 Talence (France)       http://pari.math.u-bordeaux1.fr/  [PARI/GP]
> `