Re: Decomposition of ideals in hard case.

[Xavier Roblot <xavier@east308.net>]

On Mon, 2003-09-08 at 18:26, Bill Allombert wrote:

> I would like to compute the decomposition of a prime number
> in a number field, in hard case.
> 
> Hard case mean the prime number divide the index of the defining
> polynomial, I don't have the nfinit() around, and probably I cannot
> factor the discriminant of the polynomial.
> 
> I use the code below:
>
> decomp(P,p)=
> {
>   d=poldisc(P);
>   B=nfbasis(P,,Mat([p,valuation(d,p)]));
>   /*B is a basis of a p-maximal order*/
>   K=nfinit([G.pol,B],1);
>   idealprimedec(K,p)
> }
> 
> As an example try P=x^54 + 796392*x^36 + 292918032*x^18 + 1259712.
> I don't neccessarily need all the info given by idealprimedec.
> The ramification and residual degrees can be enough in some case.
> 
> Can it be done faster? The call to nfinit() seems to waste time
> doing too much things.

Indeed, since you start by computing a p-maximal order with nfbasis
which uses Round4, you already have computed all the information
necessary to get the prime ideal factorization (see
http://almira.math.u-bordeaux.fr/jtnb/2002-1/Fordetal.ps). On the other
hand, the necessary code to make use of this information is lacking so
you'll probably need a bit of hacking of the source code to make it work
directly. But, if you only need the ramification index and residual
degree, it's a lot easier since the information is computed and actually
printed by nilord if your debug level is big enough. Anyway, I think it
should be possible without too much work to write a function that would
return, given an irreducible polynomial and a prime number p, a
p-maximal order basis for the corresponding number field and the prime
decomposition of p according to this basis.

Xavier