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. _______________________________________________ Forum mailing list [email protected] http://mail.gap-system.org/mailman/listinfo/forum
