The GAP-package is very good in working with finite fields or Abelian extensions of the Rational numbers, but sooner or later we will need to use the coordinate ring or function field of an affine variety for which it is hopeless. On the other hand, there is an excellent free package to do these calculations : Singular. So, the ideal situation for us would be to be able to access Singular from within GAP. Fortunately, Marco Costantini and Willem de Graaf have written such an interface. Here is how to get in working under OS X : One has to download two files from the Singular Mac OS X download page : Singular-2-0-4-ppcMac-darwin.tar.gz and Singular-2-0-4-share.tar.gz. Once they are on your desktop you can follow the instructions on the INSTALL.html file in the 2-0-4 Folder of the expanded Singular-2-0-4-ppcMac-darwin. Keep the tarred version and open the INSTALL-file in your browser (to be able to copy and paste) and open up the Terminal. Do the analog thing to 

cd /usr/local sudo tar -pxf
/Users/lieven/Desktop/Singular-2-0-4-ppcMac-darwin.tar sudo tar -pxf
/Users/lieven/Desktop/Singular-2-0-4-share.tar

Then follow the instructions making the symbolic links and you have Singular working. The next step is to go to the GAP Packages page and go to the package Singular for full documentation. To use Singular in a GAP-session, here is an example
gap>
LoadPackage("singular"); 
The GAP interface to Singular 
true 
gap> StartSingular();
I  Started Singular (version 2004) 
gap> SetInfoLevel( InfoSingular, 2 ); 
gap> G:= SymmetricGroup( 3 );; 
gap> R:= PolynomialRing( GF(2), 3 );; 
gap> GeneratorsOfInvariantRing( R, G ); 
[ x1 x2 x3, x1x2 x1x3 x2x3, x1x2*x3 ] 
gap> I:= Ideal( R, last );; 
gap>GroebnerBasis( I );
I  running GroebnerBasis... I  done
GroebnerBasis. [ x1 x2 x3, x2^2 x2*x3 x3^2, x3^3 ] 
gap>

mac, OSX, PPC