Re: Splitting fields

[Bill Allombert <allomber@math.u-bordeaux.fr>]

On Tue, Dec 13, 2005 at 10:16:48AM +0100, Jeroen Demeyer wrote:
> Hello list,
> 
> Suppose I have given an irreducible polynomial f(x) in Z[x].
> I want to use PARI/GP to find the splitting field (as a nf) of this 
> polynomial, i.e. the smallest field containing *all* roots of f(x).
> 
> The problem is that Q[x]/f(x) is not equal to the splitting field if 
> it's not Galois.  For instance, the splitting field of x^3 - 2 has degree 6.
> 
> Is there a way to do this?

Sure, but beware that the splitting field is generally of very large
degree hence not practical to use.

There are several way to compute it; The simplest is to use
polcompositum, for example:

split(f)=
{ 
  local(V,F1,F2);
  F1=f;F2=0;
  until(poldegree(F1)==poldegree(F2),
    F2=F1;
    V=polcompositum(F1,f);F1=V[#V]);
  F1
}

Cheers,
Bill.