Good question. If you define the ideal first by I=K.ideal(-1-3*w) and then type I.residue_symbol? to see the documentation for the function, you see that it clearly says that the output is a root of unity in the firld K itself, so is either w or w^-1 = -w-1. Which of the complex cube roots of unity is it? That is not a well-defined question since it depends on which embedding of K into CC you choose; the computation is entirely algebraic.
sage: [phi(w) for phi in K.embeddings(CC)] [-0.500000000000000 - 0.866025403784439*I, -0.500000000000000 + 0.866025403784439*I] If you want w to embed to the first of these, you can specify that when you create K: sage: K.<w>=NumberField(x^2+x+1, embedding=CC(-0.5,0.866)) sage: CC(w) -0.500000000000000 + 0.866025403784439*I (the precision of the root which you give is not important as long as it is sufficient to distinguish one root from the others). Now you can say sage: K.ideal(-1-3*w).residue_symbol(5,3) -w - 1 sage: CC(K.ideal(-1-3*w).residue_symbol(5,3)) -0.500000000000000 - 0.866025403784439*I John Cremona On 2 April 2014 12:25, <[email protected]> wrote: > Hi, I created a numberfield: > K.<w>=NumberField(x^2+x+1) > Then I wanted to know the following residue symbol: > K.ideal(-1-3*w).residue_symbol(5,3) > Sage said the answer is: > -w-1 > > My problem is: What is meant by the element w? > Does it mean w=e^(2pi*i/3) or is it w=e^(2pi*i/3)^2? > Which one of the roots of unity does sage use as w? I dont know, how to find > it out. It is really important for me, so it would be great if anyone could > help me. > I cannot use the forum, because I am not allowed to ask questions. > Please help me! > > -- > You received this message because you are subscribed to the Google Groups > "sage-support" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/sage-support. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.
