[GAP Forum] working with GroupRings

[tkohl at math.bu.edu]

Dear Forum members,

This is somewhat related to a question I asked a while
back about GroupRings. My question is somewhat general,
but I will try to be as brief as possible.

I am trying to construct the group ring  Q[w,z]S_3 where
Q[w,z] is the field extension of Q obtained by adjoining w,z where
w^3=2 and z is a primitive cube root of unity.

The method I am using is this:

z:=Indeterminate(Rationals,"z");
w:=Indeterminate(Rationals,"w");
R:=PolynomialRing(Rationals,["z","w"]);
I:=Ideal(R,[z^2+z+1,w^3-2]);
F:=R/I;
S3:=SymmetricGroup(3)
FS3:=GroupRing(F,S3);

so far so good.

One initial thing I notice is this:

gap> BF:=BasisVectors(Basis(F));
[ (1), (w), (w2), (z), (zw), (zw2) ]

which I can understand corresponds to the ideals 1+I, w+I, w^2+I, etc.
but I am not sure how to actually construct expressions by hand.

[That (zw2) is the representative instead of (z*w^2) is a bit jarring too, but that's
not the biggest issue.]

i.e. This does not work

gap> (w) in F;
false

although if I do 

gap> BF[2] in F

then, of course, it is true. 

Q1) How can I specify elements of F without having to refer to the literal list returned
from BasisVectors(Basis(F))  ?

Once I'm past this hurdle, I still want to work with elements of FS3 by taking linear
combinations of group elements and elements of F. 

Q2) I want to be able to do something like this:

(z*(1,2))*(z*(1,2)) 

and have it give me (-z-1)*()

I know I need to use One(F) or One(FS3) in these expressions, but everything I have tried
ends up triggering

"Error, no method found! For debugging hints type ?Recovery from NoMethodFound"

Q3) Alternately, is there a way (like in Maple) to symbolically manipulate a polynomial
expression, for example

algsubs(z^2+z+1=0,z^4+z5)

and yield z+z^2? 

(i.e. Forget about using a quotient ring and instead apply some regular expression
to 'mod out' by the relations w^3=2 and z^2+z+1=0.)

Pardon the length of my question, and thanks in advance for any assistance.
The main reason I'm using GAP in this instance is that Maple's grouptheory and
non-commuting variables infrastructure didn't work.

Thanks.

	-Tim K.