Re: A number fields question

[Bill Allombert <Bill.Allombert@math.u-bordeaux1.fr>]

On Wed, Aug 19, 2009 at 10:49:47AM +0100, Slessor R. wrote:
> 
> Dear all,
> 
> I was wondering if you would be able to help me with a quick question? I have the following relative number fields extension L/K (also Galois):
> 
> K=nfinit(y^2 + y + 1);
> f= x^6 + Mod(3969*y+10584,y^2+y+1);
> L=rnfinit(K,f);
> lambda = Mod(Mod(-4/64827*y-1/43218,y^2+y+1)*x^5 + Mod(2/9261*y-1/9261,y^2+y+1)*x^4 + Mod(-1/294*y-1/441,y^2+y+1)*x^3 + Mod(1/441*y+1/147,y^2+y+1)*x^2 + Mod(-1/42*y-1/42,y^2+y+1)*x+1/3, x^6 + Mod(3969*y+10584,y^2+y+1));
> 
> 
> lambda has been chosen in such a way that it is totally real under the action of Gal(L/K)
> 
> I would like to know if there is a way to get PARI to compute all elements x of the ring of integers O_L of L such that |Tr_{L/K}(lambda. \bar{x}. x)| = 1?
> 
> where \bar{x} is just the usual complex conjugation.

I would try the following:
Compute an integral basis for O_L and consider
q(x)=Tr_{L/K}(lambda. \bar{x}. x) as a quadratic form over O_L 
with value in C and compute its matrix. then you should be able
to answer your question.

Cheers,
Bill.