[GAP Forum] Finding a polynomial whose Galois group is M12

[Bill Allombert]

On Wed, Sep 13, 2017 at 12:02:37PM +0000, johnathon simons wrote:
> Dear all,
> 
> Hope the the remainder of the Summer is treating you well. I've a brief question:
> 
> Once one has realized a finite group G (say the Mathieu group M12) as
> a Galois group over Q, is there any method on how to extract a
> polynomial whose Galois Group is M12? I've seen some papers where they
> state the polynomials over Q  whose Galois group is M12, but I've
> never actually seen anything method of doing so. If so, is such an
> implementation possible on GAP.

How do you give the realization ?

In general, even when one has built a field with the right Galois
group, obtaining a polynomial can be computationaly challenging,
either due to its size of the polynomial or of intermediary computations.

For example it is easy to find a field with Galois group 31:3.
However computing a corresponding deg-31 polynomial is much harder,
so you will have to add extra conditions on the field so that the
computation is manageable.

Cheers,
Bill.