Bill Allombert on Sat, 26 Aug 2006 23:27:01 +0200

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Re: Extensions of abelian Numberfields...

On Sat, Aug 26, 2006 at 02:21:15AM +0200, wrote:
> Hello,
> i am an absolute PARI/GP beginner, so may be my question is trivial: How does
> PARI/GP deal with Extensions
> IQ C K C K' C IQ(z) with a root of unity z? Especially: Are there examples for
> computing norms and traces for K'/K ? Thank you for every answer.

(Please always consider including an detailed example of what you want to
compute, this saves time.)

Suppose K=Q(sqrt(-2)) and K'=Q(sqrt(1-2*sqrt(-2)))

We define a=sqrt(-2) by identifing Q(sqrt(-2)) and Q[Y]/(Y^2+2):

? a=Mod(y,y^2+2)

We define b=sqrt(1-2*a) by identifing K(sqrt(1-2*a)) and

? b=Mod(x,x^2-(1-2*a))

Then we can use the GP functions norm() and trace():
? trace(b)
%3 = 0
So trace_K'/K(b)=0

? norm(b)
%4 = Mod(2*y - 1, y^2 + 2)
So norm_K'/K(b)=2*sqrt(-2)-1

Note that this does not require the extension K/K' to be abelian.