Re: A question on galois extensions in Pari/GP

[mehnert@math.uni-sb.de]

Hello,

i have indeed an older Pari/GP version, namely version 2.1.6-3. So i have to
install Pari/GP 2.3 .The reason why i use the older version is that i work with
an debian sarge Linux distribution and i didn't find a backport of Pari/GP 2.3
for the sarge.

Regards,
Bernd



Quoting 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.
>




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