Georgi Guninski on Sat, 14 Jul 2012 09:56:26 +0200 |
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Potential inconsistent results with fundamental units and KASH 3 |
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.