Markus Endres on 27 Feb 2003 19:25:48 +0100

 [Fwd: Re: nfgaloisconj]

```-----Forwarded Message-----

> From: Bill Allombert <allomber@math.u-bordeaux.fr>
> To: pari-users@list.cr.yp.to
> Subject: Re: nfgaloisconj
> Date: 27 Feb 2003 16:24:58 +0100
>
> 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 ?

of course, here it is:

? K=bnfinit(y^2+7);

%7 = x^4 + Mod(-y, y^2 + 7)*x^3 - 3*x^2 + Mod(y, y^2 + 7)*x + 1

? rnfequation(K,%)
%8 = x^8 + x^6 - 3*x^4 + x^2 + 1

? L=bnfinit(%);

? aut=nfgaloisconj(L)
%10 = [x, 1/2*x^7 - 1/2*x^6 + x^5 - 1/2*x^4 - 1/2*x^3 + x^2 - 1/2, x^7 + x^5 - 3*x^3 + x,
1/2*x^7 + 1/2*x^6 + x^5 + 1/2*x^4 - 1/2*x^3 - x^2 + 1/2, -1/2*x^7 - 1/2*x^6 - x^5 - 1/2*x^4 + 1/2*x^3 + x^2 - 1/2,
-x^7 - x^5 + 3*x^3 - x, -1/2*x^7 + 1/2*x^6 - x^5 + 1/2*x^4 + 1/2*x^3 - x^2 + 1/2, -x]~

now I need all the automorphisms in aut which leaves the elements of K fix, i.e. gal(L|K).
but these automorphisms are defined absolute over Q, and I need them relative over K

thx

markus

>
> 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.
--
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