Karim BELABAS on Sun, 5 Oct 2003 15:21:50 +0200 (MEST)

 Re: Cyclotomic Integer with a Given Norm?

On Sun, 5 Oct 2003, Bill Allombert wrote:
> On Sun, Oct 05, 2003 at 03:42:36PM +0800, B.Y. wrote:
> > 	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)
[...]
> 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.

A closer alternative is

bnfisintnorm(B, p)

which computes a complete system of solutions (mod units) of the equation
Nx = p, with x an algebraic integer.

Karim.
--
Karim Belabas                     Tel: (+33) (0)1 69 15 57 48
Dép. de Mathématiques, Bât. 425   Fax: (+33) (0)1 69 15 60 19
Université Paris-Sud              http://www.math.u-psud.fr/~belabas/
F-91405 Orsay (France)            http://www.parigp-home.de/  [PARI/GP]