Karim Belabas on Sat, 14 Jul 2012 11:03:58 +0200

 Re: Potential inconsistent results with fundamental units and KASH 3

```* 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]
`

```