Dear GAP-Forum,

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,

Alexander Hulpke

Thanks for the reply, I was looking for a (relatively) simple command to
calculate the galois groups in either GAP3 or GAP4, rather than reducing the
polynomials modulo certain unramified primes and trying to determine the
galois group from the cycle structures of some its elements. I tried using
the GaloisGroup(F); command in GAP3, where F is the splitting field of the
polynomial but I was unable to define the splitting field properly in GAP3.
However, Igor Schein gave me a simple answer:

Using the software KANT 2.0 (Kalculations in Algebraic Number Theory), which
is very similar to GAP, type the commands:

Galois(x^12-12*x+1);
Galois(11*x^12-12/11*x^11+1);

KANT 2.0 can determine galois groups of polynmials up to degree 15. Can this
be done in GAP3 in a relatively easy way?

David.
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