Bill Allombert on Thu, 27 Feb 2003 16:24:58 +0100

 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.
```

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