[GAP Forum] Galois Group question

[Bill Allombert]

On Mon, May 15, 2006 at 02:19:15PM -0400, Igor Schein wrote:
> Dear Gap Forum,
> 
> Given a small group G of order 2^n, I would like to know whether or not 
> there exists a polynomial P in x^4 whose Galois group is G.  Let me
> illustrate:
> 
> gap> IdGroup(TransitiveGroup(8,GaloisType(x^8+3*x^4+1)));
> [ 8, 3 ]
> 
> So the answer is yes for small group [8,3],  P: x -> x^2+3*x+1
> 
> However, if I consider small group [8,4], such P clearly doesn't
> exist, so the answer is no.
> 
> So my question is how I can answer this question using GAP commands
> and intrinsic properties of groups.  Specifically, I need to know the
> answer for [64,64] and [64,122].

I don't think your question can be easily expressed in a purely group
theoretic way, because the obstruction is arithmetic in nature.

Let us show one of easiest arithmetic obstructions:

Let K be the normal closure of the splitting field of P.
Then K must include the 4th roots of unity. 

However the field Q(zeta_4)=Q(sqrt(-1) cannot be extended to a C4
extension of Q because -1 is not the sum of two squares in Q.

We can extend this result using group theory by saying that the
abelianized of G must have at least one abelian invariant not divisible
by 4, (since G must have a C2 quotient (corresponding to Q(sqrt(-1) through
Galois theory) that cannot be lifted to a C4 quotient) but the basis
of the obstruction is arithmetic.

While this is the only obstruction for abelian groups of order divisible
by 4, there are other obstructions for non-abelian groups.

Cheers,
Bill.