Xavier Roblot on Tue, 09 Sep 2003 15:25:37 +0200 |
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Re: Decomposition of ideals in hard case. |
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