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. _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum