I would say the second one is correct:
(%i6) divide(x^5+1,2*x^2+(-sqrt(5)-1)*x+2);
(%o6) [(8*x^3+(4*sqrt(5)+4)*x^2+(4*sqrt(5)+4)*x+8)/16,0]
(%i7) expand((2*x^2+(-sqrt(5)-1)*x+2)*%[1]);
(%o7) x^5+1
Andrej
On 5/13/06, Raymond Toy <raymond.toy at ericsson.com> wrote:
>
> I'm not very familiar with algebraic integers so I hope someone can
> tell if the following is correct or not. It appears in the middle of
> a calculation of Andrej's example of
> ratsimp(diff(integrate(1/(x^5+1),x),x)).
>
> (%i42) display2d:false;
> (%o42) false
> (%i43) algebraic:false;
> (%o43) false
> (%i44) gcd(x^5+1,2*x^2+(-sqrt(5)-1)*x+2,x);
> (%o44) 1
>
>
> (%i45) algebraic:true;
> (%o45) true
> (%i46) gcd(x^5+1,2*x^2+(-sqrt(5)-1)*x+2,x);
> (%o46) 2*x^2+(-sqrt(5)-1)*x+2
>
> The first gcd makes sense to me. But is the second gcd correct when
> algebraic is true?
>
> Ray
>
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