Re: A question on galois extensions in Pari/GP

[Bill Allombert <allomber@math.u-bordeaux.fr>]

On Mon, Sep 04, 2006 at 10:37:08AM +0200, mehnert@math.uni-sb.de wrote:
> Hello,
> 
> I have question on the issue "galois extensions":
> Regard the following computation:
> 
> ? F=galoisinit(polcyclo(28));
> 
> ? K=galoisfixedfield(F,F.gen[2],2)
> %36 = [x^6 - 12*x^5 + 53*x^4 - 104*x^3 + 86*x^2 - 24*x + 1, Mod(-x^11 + x^9 -
> x^7 + x^5 - x^3 + 2*x + 2, x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1), [x^2 +
> (1/28*y - 1/14)*x - 1/28, x^2 + (1/28*y^5 - 3/49*y^4 + 13/196*y^3 - 1/98*y^2 +
> 1/196*y - 1/14)*x - 1/28, x^2 + (1/28*y^3 - 13/196*y^2 + 5/196*y - 1/14)*x -
> 1/28, x^2 + (-1/28*y + 1/14)*x - 1/28, x^2 + (-1/28*y^3 + 13/196*y^2 - 5/196*y
> + 1/14)*x - 1/28, x^2 + (-1/28*y^5 + 3/49*y^4 - 13/196*y^3 + 1/98*y^2 - 1/196*y
> + 1/14)*x - 1/28]]
> 
> ? a=Mod(u,subst(K[1],x,u))
> %37 = Mod(u, u^6 - 12*u^5 + 53*u^4 - 104*u^3 + 86*u^2 - 24*u + 1)
> 
> ? prod(i=1,6,subst(K[3][i],y,a))
> 
> Shouldn't this produce F[1], i.e.:
> 
> ? F[1]
> %39 = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1

Yes, it should, and it does with PARI/GP 2.3.0

F=galoisinit(polcyclo(28)); K=galoisfixedfield(F,F.gen[2],2); a=Mod(u,subst(K[1],x,u)); lift(prod(i=1,6,subst(K[3][i],y,a)))
%4 = x^12 + Mod(-1, u^6 - 2*u^5 + 4*u^4 - 8*u^3 + 16*u^2 - 32*u + 64)*x^10 + Mod(1, u^6 - 2*u^5 + 4*u^4 - 8*u^3 + 16*u^2 - 32*u + 64)*x^8 + Mod(-1, u^6 - 2*u^5
+ 4*u^4 - 8*u^3 + 16*u^2 - 32*u + 64)*x^6 + Mod(1, u^6 - 2*u^5 + 4*u^4 - 8*u^3 + 16*u^2 - 32*u + 64)*x^4 + Mod(-1, u^6 - 2*u^5 + 4*u^4 - 8*u^3 + 16*u^2 - 32*u + 64)*x^2 + Mod(1, u^6 - 2*u^5 + 4*u^4 - 8*u^3 + 16*u^2 - 32*u + 64)
? lift(%)
%5 = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1

> Where is the error here? Thank you for any answer.

I expect you are using an old PARI version, probably 2.1.x, which is
known to have this problem (Bug 2.2.1F21, which could be described by
"galoisfixedfield(,,2) is broken and frequently returns garbage under
PARI version < 2.2.1)").

In that case I suggest you upgrade to PARI/GP 2.3.0, the current stable
release.

Cheers,
Bill.