David Burggraf asked:
I'm just wondering if anyone know if it's possible for GAP4 to compute the
galois groups of the following polynomials over the rationals:x^12-12*x+1 x^12-12/11*x^11+1
If so, an explanation of how it is done would be greatly appreciated.
GAP4 unfortunately does not yet have the functionality to recognize Galois
groups. (The functionality exists in GAP3 via the `Galois' command. This
will be ported to GAP4 at a later point.)
However in this particular case one can prove that both polynomials have
Galois group S12 also in GAP4 by factoring modulo different primes, which
don't divide the discriminant.
I can give you more details if you want (I'm not sure from your question
whether you need a command to do it or whether you want an explanation of
the mathematics involved).
I hope this helps,