Dear Forum, concerning the question asked by Tim Kohl, about dealing with certain subspaces of group algebras, I am not sure whether I understand what the goal is.

Perhaps we try a different approach, using the implementation of cyclotomic fields instead of the algebraic extensions. (Of course this is neither general enough nor practically desirable.) Consider the following GAP session. gap> r2:= Sqrt( 2 );; # a square root of 2 gap> K:= Field( Rationals, [ r2 ] );; # the field extension gap> S3:= SymmetricGroup( 3 );; gap> KS3:= GroupRing( K, S3 );; gap> emb:= Embedding( S3, KS3 );; gap> gens:= [ (1,2)^emb + (1,2,3)^emb, (1,3,2)^emb + (1,3)^emb ]; [ (1)*(1,2)+(1)*(1,2,3), (1)*(1,3,2)+(1)*(1,3) ] Now one wants to create a Q-space that is generated by some elements in KS3. For that, we can either view the group algebra as a Q-space, ... gap> Q_KS3:= AsAlgebra( Rationals, KS3 );; gap> Dimension( Q_KS3 ); 12 gap> Dimension( KS3 ); 6 gap> H1:= Subspace( Q_KS3, gens ); <vector space over Rationals, with 2 generators> ... or we form the vector space independent of the group algebra, just by prescribing the base field and generators. (The two variants are of course equal as sets.) gap> H2:= VectorSpace( Rationals, gens ); <vector space over Rationals, with 2 generators> gap> H1 = H2; true In both cases, forming products of elements in the subspaces works. gap> prod:= gens[1] * gens[2]; (1)*()+(1)*(2,3)+(1)*(1,2)+(1)*(1,2,3) gap> prod in H; false Is this roughly the setup of interest? If yes then the analogous construction using general algebraic extensions would require to deal with algebras/spaces over subfields of the extension. Is that available? All the best, Thomas On Thu, Oct 26, 2017 at 11:15:16AM -0400, tk...@math.bu.edu wrote: > > Dear forum, > > This has gotten me part way to what I'm looking for. > > (Many thanks Frank.) > > But I'm running into a different problem now. > > Basically, if one has, for example, > > a:=Indeterminate(Rationals,"r"); > K:=AlgebraicExtension(Rationals,r^2-2) > S3:=SymmetricGroup(3); > KS3:=GroupRing(K, S3); > emb:=Embedding(S3,KS3); > > then I would like to be able to view KS3 as > a module over Q, so that I can do something > like this > > H:=Subspace(KD3,[(1,2)^emb+(1,2,3)^emb, (1,3,2)^emb+(1,3)^emb ]); > > so that H is the Q-span of { (1,2)+(1,2,3) , (1,3,2)+(1,3) } > with the ultimate goal of being able to multiply elements > of H and represent them with respect to this basis. > > [I'm basically looking at Q-subalgebras of KG.] > > Also, as an aside, the GaloisGroup() function seems > not to be working. It gives the "no method found" error > if I try to do GaloisGroup(K). [I'm using 4r8.] > > Thanks. > > -T _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum