|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, firstname.lastname@example.org 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 K[X]/(X^2-(1-2*a)): ? 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. Cheers, Bill.