Bill Allombert on Thu, 27 Feb 2003 16:24:58 +0100 |
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Re: nfgaloisconj |
On Thu, Feb 27, 2003 at 03:06:37PM +0100, markus endres wrote: > hi > > assume L|K|Q is a tower of fields (Q the rationals). > L|K a galois extension. > > then nfgaloisconj(L) determines the automorphisms from L over Q, hence > the galois group gal(L|Q) contains these automorphisms defined over Q. > > but now, I want to compute the galois group gal(L|K).(ok, this is easy. > I look at the automorphisms of gal(L|Q) which fixes K pointwise). > > now, I have gal(L|K) with automorphisms defined over Q, but I need these > automorphisms defined over K. > > How can I do this? Could you send us a practical computation you want to perform ? Anyway, here is how I see the problem: Suppose L is given by a polynomial P and G=galoisinit(P); (say P=x^4+1) Suppose you know a subset H of G.group that generate gal(L|K). (say H=G.gen[1]) Now compute F=galoisfixedfield(G,H,2); ? F[3][1] %5 = x^2 - 1/2*y We convert it to a true relative polynomial with: R=F[3][1]*Mod(1,subst(F[1],x,y)) R is a relative polynomial defining K/L and have the nice property that it divides P. Now galoispermtopol(G,H)%R is the definition of H over K. Cheers, Bill.