GAP usage/function FAQ

[Jeffreyke]

I wish someone can give me a hand on following application problems:
(1) How to get quotient Ring:
E.g.

R :=Integers;
Integers
gap> S :=Ideal(R, [17]);
<two-sided ideal in Integers, (1 generators)>
gap> R_S := R/S;
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method .......

I couldn't build Quotient Ring like Quotient Group

(2) Why Factors() doesn't work as expected on GaussianIntegers Ring ?

x :=X(GaussianIntegers, "x");
gap> Factors(x^2+1);
[ x^2+1 ]

Based on Ring coefficients definition, Factors(x^2+1) should get:
[x - i, x+i]
Right?

(3) Can I extract a Group structure from some existing Ring for its multiplication group?

 gap> z79 := Integers mod 79;
GF(79)
gap> IsField(z79);
true
gap> StructureDescription(z79);
Error, no method found! For debugging hints type ? ....

How can I analyze some Ring's multiplication group independently?

[fingolfin]

Hi there!

First, a meta remark: I took the liberty of editing your post a bit, it was very difficult to read due to your user of ===== as separators, which causes text to render in BIG BOLD LETTERS -- I've replaced those by triple backticks, which exist to render code, and the result looks better.

Regarding your first question: To get Z/nZ, just write Integers mod n. For primes, though, GAP will by default return the finite field GF(p):

gap> Integers mod 17;
GF(17)
gap> Integers mod 17^2;
(Integers mod 289)

Regarding your second question: the variable x does not really store the coefficient domain. You need to pass the polynomial ring in which you want to factor to the Factors function (its manual entry explains that and gives an example). Finally, GAP really only knows how to factor over coefficient domains which are fields. So try this:

gap> R:=PolynomialRing(GaussianRationals,["x"]);
GaussianRationals[x]
gap> x:=R.1;
x
gap> Factors(R, x^2+1);
[ x+(-E(4)), x+E(4) ]

Regarding the third point:

gap> z79 := Integers mod 79;
GF(79)
gap> StructureDescription(Units(z79));
"C78"

But of course we knew that: the unit group of a finite field is always a cyclic group.