[GAP Forum] quotients in polynomial rings

[David Joyner]

The exact functionality you want is missing from GAP at the moment.
However, there may be a work around.

The following example computation seems to be related to what
you are looking for. Computations in R/I must be replaced
by repeatedly using QuotientRemainder and PowerMod.

gap> Z4:=ZmodnZ(4);
(Integers mod 4)
gap> R:=UnivariatePolynomialRing(Z4,1);
PolynomialRing(..., [ x ])
gap> x:=IndeterminatesOfPolynomialRing(R)[1];
x
gap> I:=TwoSidedIdealByGenerators( R,[x8-x0]);
<two-sided ideal in PolynomialRing(..., [ x ]), (1 generators)>
gap> gen:=x8-x0;
x8-ZmodnZObj(1,4)
gap> QuotientRemainder(R,x8,gen);
[ ZmodnZObj(1,4), ZmodnZObj(1,4) ]
gap> QuotientRemainder(R,x15,gen);
[ x7, x7 ]
gap> QuotientRemainder(R,x15+x8,gen);
[ x7+ZmodnZObj(1,4), x7+ZmodnZObj(1,4) ]
gap> PowerMod( R, x+x0, 15, gen );
ZmodnZObj(0,4)
gap> PowerMod( R, x, 15, gen );
x7

Wish I could be more helpful.

- David Joyner

++++++++++++++++++++++++++++++++++++++++++++


Larry Wilson wrote:

>Perhaps this question has a very simple answer, but I have not been able
>to figure it out and would love to have some guidance.  I would like to be
>able to do something like work in Z_4[x]/(x^8-1).  I thought I was on the
>right track with:
>
>GAP4, Version: 4.4.2 of 18-Mar-2004, sparc-sun-solaris2.8-cc
>gap> A := FreeAlgebraWithOne(Integers, 1);;
>gap> x := GeneratorsOfAlgebra(A)[2];
>(1)*x.1
>gap> I := Ideal(A, [x^8-1*x^0, 4*x^0]);
><two-sided ideal in <free left module over Integers, and ring-with-one,
>with 1 generators>, (2 generators)>
>gap> hom := NaturalHomomorphismByIdeal(A, I);
>Error, no method found! For debugging hints type ?Recovery from
>NoMethodFound
>Error, no 1st choice method found for
>`GeneratorsOfLeftOperatorRingWithOne' on 1 arguments called from
>GeneratorsOfAlgebraWithOne( image ) called from
><function>( <arguments> ) called from read-eval-loop
>
>Any help would be much appreciated,
>
>Larry
>
>
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