Bill Allombert on Sun, 5 Oct 2003 14:37:19 +0200

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Re: Cyclotomic Integer with a Given Norm?

On Sun, Oct 05, 2003 at 03:42:36PM +0800, B.Y. wrote:
> Dear Sirs (and Mesdames):
> 	Can someone tell me if there is a ready and easy way to decide 
> with PARI whether there exists an integer $a$ in a cyclotomic field 
> $Q(\zeta_n)$ with a given norm $p$, where $p$ is a prime number?

If I am not mistaken, your problem is equivalent to deciding whether
1) the ideal above p are of norm p and 2) the ideals above p are principals.

You can do:

? B=bnfinit(polcyclo(n));
? L=idealprimedec(B,p)[1];
? M=bnfisprincipal(B,L)

p is the norm of an integer of B iff L.f==1 and M[1]==0, the integer being M[2]
(expressed on the integral basis).

The above work because Q(\zeta_n) is Galois and totally complex (for n>2).

An alternative solution that look simpler but may not do what you want:

? N=bnfisnorm(B,p,0)

p is a norm of a element N[1] of B iff N[2]==1 under the GRH, but
N[1] is not warranted to be integral.