Re: How to determine Mod(a,b) with t_COMPLEX b?

On Tue, May 27, 2025 at 12:53:44PM +0200, [email protected] wrote:
> On 2025-05-27 00:40, Bill Allombert wrote:
> > > 
> > > The minimal residue of 1+4*I modulo 3+2*I is the yellow point -I in
> > > the
> > > example:
> > > https://en.wikipedia.org/wiki/Gaussian_integer#Describing_residue_classes
> > > 
> > > How can minimal residue of an input gaussian integer modulo a gaussian
> > > integer be computed in PARI/GP?
> > 
> > nf=nfinit(i^2+1)
> > a=1+4*i;b=3+2*i;
> > nfeltdivrem(nf,a,b)
> > %4 = [[1,1]~,[0,-1]~]
> > 
> > Cheers,
> > Bill.
> > 
> Thank you, so the rem part is what I asked for.
> Interesting, "i" is free variable and not sqrt(-1) which is "I" in GP.
> 
> Now that type(b) is t_POL, Mod(a,b) works as well.
> What is the meaning of "-5" in Mod(a,b) result?

This is the remainder of the Euclidean division of 1+4*x by 3+2*x.
Indeed:

? divrem( 1+4*x , 3+2*x)
%1 = [2,-5]~

Cheers,
Bill.