mehnert on Mon, 04 Sep 2006 10:44:25 +0200

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 A question on galois extensions in Pari/GP

```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))
%38 = x^12 + Mod(-199665/19208*u^5 + 2921083/38416*u^4 - 3603743/19208*u^3 +
415570/2401*u^2 - 121510/2401*u + 72857/38416, u^6 - 12*u^5 + 53*u^4 - 104*u^3
+ 86*u^2 - 24*u + 1)*x^10 + Mod(1566191951/46118408*u^5 -
98254156643/368947264*u^4 + 508163823659/737894528*u^3 -
961316890439/1475789056*u^2 + 142099993417/737894528*u - 1490999831/184473632,
u^6 - 12*u^5 + 53*u^4 - 104*u^3 + 86*u^2 - 24*u + 1)*x^8 +
Mod(-1494368708311/578509309952*u^5 + 11737813222581/578509309952*u^4 -
7596782692955/144627327488*u^3 + 57518013549907/1157018619904*u^2 -
75931597927/5165261696*u + 357291948815/578509309952, u^6 - 12*u^5 + 53*u^4 -
104*u^3 + 86*u^2 - 24*u + 1)*x^6 + Mod(1566191951/36156831872*u^5 -
98254156643/289254654976*u^4 + 508163823659/578509309952*u^3 -
961316890439/1157018619904*u^2 + 142099993417/578509309952*u -
1490999831/144627327488, u^6 - 12*u^5 + 53*u^4 - 104*u^3 + 86*u^2 - 24*u +
1)*x^4 + Mod(-199665/11806312448*u^5 + 2921083/23612624896*u^4 -
3603743/11806312448*u^3 + 207785/737894528*u^2 - 60755/737894528*u +
72857/23612624896, u^6 - 12*u^5 + 53*u^4 - 104*u^3 + 86*u^2 - 24*u + 1)*x^2 +
1/481890304

Shouldn't this produce F[1], i.e.:

? F[1]
%39 = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1

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

Regards,
Bernd

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