Georgi Guninski on Sat, 14 Jul 2012 09:56:26 +0200

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 Potential inconsistent results with fundamental units and KASH 3

• To: pari-dev@pari.math.u-bordeaux.fr
• Subject: Potential inconsistent results with fundamental units and KASH 3
• From: Georgi Guninski <guninski@guninski.com>
• Date: Sat, 14 Jul 2012 10:56:20 +0300
• Delivery-date: Sat, 14 Jul 2012 09:56:36 +0200
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• User-agent: Mutt/1.5.20 (2009-06-14)

```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?

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

Can't solve either for n<=10^4 and the coefficient are growing quite fast.

I suspect if a solution doesn't exist then rank will be greater than 1.

What am I missing?

Thanks.

```