[GAP Forum] Sqrt(1+2i)

[Marek Mitros]

Dear Forum Users,

I need to calculate Sqrt(1+2i) i.e. find the complex number z=a+bi
such that z^2=(1+2i). Using traditional pen and paper I calculated
that a=Sqrt((1+Sqrt(5))/2) and b=Sqrt((Sqrt(5)-1)/2). But how to
express these numbers in GAP ? Is this number cyclotomic or not ?
Using formula for tangent (x/2) = (1-cos(x))/sin(x) I obtain number
c=1+ ((Sqrt(5)-1)/2)*i which is collinear with needed number i.e. have
the same angle. So we have ImaginaryPart(c*(1-2*i))=0.

Another question I have is how to normalize complex number in GAP.
E.g. I have number c=1+ ((Sqrt(5)-1)/2)*i and I would like to find
number c/|c| i.e. lying on unit circle on complex plane. If the |c|^2
is rational then I can apply Sqrt. But this does not work for real
cyclotomics.

Any advice ?

Regards,
Marek