############################### #This file contains a series of #adjustments to the program #finding normal elements to find a #normalizing sequence for the algebra F(1, \gamma). #The output is reproduced at the end of the file. #The first part #of the program below looks for normal elements of degree 3 in #F(1, \gamma). ############################ rr1d4_sub_list:={seq(seq(x[r,s,2,1]= p*x[r,s,1,2], r = 1..3), s =1..3)} union {seq(seq(x[r,2,1,s]= p*x[r,1,2,s], r = 1..3), s =1..3)} union {seq(seq(x[2,1,r,s]= p*x[1,2,r,s], r = 1..3), s =1..3)}; rr2d4_sub_list:={seq(seq(x[r,s,3,2] = a*x[r,s,3,1] +x[r,s,1,3] + b*x[r,s,2,3], r =1..3), s = 1..3)} union {seq(seq(x[r,3,2,s] = a*x[r,3,1,s] + x[r,1,3,s] + b*x[r,2,3,s], r =1..3), s = 1..3)} union {seq(seq(x[3,2,r,s] = a*x[3,1,r,s] +x[1,3,r,s] + b*x[2,3,r,s], r = 1..3), s = 1..3)}; rr3d4_sub_list:={seq(x[r,3,3,1] = c*x[r,1,3,3] +d*x[r,2,3,3]+e*x[r,3,1,3], r = 1..3)} union {seq(x[3,3,1,r] = c*x[1,3,3,r] +d*x[2,3,3,r]+ e*x[3,1,3,r], r = 1..3)}; rr4d4_sub_list:= {seq(x[r,3,1,1] = f*x[r,1,1,3] + g*x[r,1,2,3] +h*x[r,2,2,3] + j*x[r,1,3,1] + k*x[r,2,3,1], r = 1..3)} union {seq(x[3,1,1,r] = f*x[1,1,3,r] +g*x[1,2,3,r] + h*x[2,2,3,r] +j*x[1,3,1,r] + k*x[2,3,1,r], r = 1..3)}; rr5d4_sub_list:= {seq(x[r,3,1,2] = q*x[r,1,3,1] + q*b*x[r,2,3,1] +a*q*x[r,3,1,1], r = 1..3)} union {seq(x[3,1,2,r] = q*x[1,3,1,r] + q*b*x[2,3,1,r] + a*q*x[3,1,1,r], r = 1..3)}; sol_sub_seq:=[{e = -1, h = 0, f = 0, j = 0, c = 0, q = RootOf(_Z^2+_Z+1), p = -1-RootOf(_Z^2+_Z+1), a = -RootOf(_Z^2+_Z+1), d = -1-RootOf(_Z^2+_Z+1), b = 1, g = RootOf(_Z^2+_Z+1), k = -1-RootOf(_Z^2+_Z+1)}]; for m from 1 by 1 to nops(sol_sub_seq) do gone_sub_list3[m]:={seq(YY[m,r,3,2]=0, r=1..3)} union {seq(YY[m,3,2,r]=0, r=1..3)} union {seq(YY[m,r,2,1]=0, r=1..3)} union {seq(YY[m,2,1,r]=0, r=1..3)} union {YY[m,3,3,1]=0, YY[m,3,1,1]=0, YY[m,3,1,2]=0}; r1d4_sub_list[m]:=subs(sol_sub_seq[m], rr1d4_sub_list); r2d4_sub_list[m]:=subs(sol_sub_seq[m], rr2d4_sub_list); r3d4_sub_list[m]:=subs(sol_sub_seq[m], rr3d4_sub_list); r4d4_sub_list[m]:=subs(sol_sub_seq[m], rr4d4_sub_list); r5d4_sub_list[m]:=subs(sol_sub_seq[m], rr5d4_sub_list); for r from 1 by 1 to 3 do PPP[m,r]:=add(add(add(YY[m,t,u,w]*x[r,t,u,w], u=1..3), t=1..3), w=1..3) end do; for r from 1 by 1 to 3 do QQQ[m,r]:=add(add(add(add(YY[m,t,u,w]*WW[m,v,r]*x[t,u,w,v], u=1..3), t=1..3), v=1..3), w=1..3) end do; for r from 1 by 1 to 3 do PPP1[m,r]:=subs(gone_sub_list3[m], PPP[m,r]) end do; for r from 1 by 1 to 3 do QQQ1[m,r]:=subs(gone_sub_list3[m], QQQ[m,r]) end do; for r from 1 by 1 to 3 do RRR1[m,r]:=PPP1[m,r]-QQQ1[m,r] end do; for r from 1 by 1 to 3 do RRR2[m,r]:=subs(r5d4_sub_list[m], subs(r4d4_sub_list[m], subs(r3d4_sub_list[m], subs(r2d4_sub_list[m],subs(r1d4_sub_list[m], RRR1[m,r]))))) end do; for r from 1 by 1 to 3 do RRR3[m,r]:=subs(r5d4_sub_list[m], subs(r4d4_sub_list[m], subs(r3d4_sub_list[m], subs(r2d4_sub_list[m],subs(r1d4_sub_list[m], RRR2[m,r]))))) end do; for r from 1 by 1 to 3 do RRR4[m,r]:=subs(r5d4_sub_list[m], subs(r4d4_sub_list[m], subs(r3d4_sub_list[m], subs(r2d4_sub_list[m],subs(r1d4_sub_list[m], RRR3[m,r]))))) end do; for r from 1 by 1 to 3 do RRR5[m,r]:=subs(r5d4_sub_list[m], subs(r4d4_sub_list[m], subs(r3d4_sub_list[m], subs(r2d4_sub_list[m],subs(r1d4_sub_list[m], RRR4[m,r]))))) end do; for r from 1 by 1 to 3 do RRR[m,r]:=subs(r5d4_sub_list[m], subs(r4d4_sub_list[m], subs(r3d4_sub_list[m], subs(r2d4_sub_list[m],subs(r1d4_sub_list[m], RRR5[m,r]))))) end do; YYlist[m]:={seq(seq(seq(YY[m,t,u,w], t=1..3), u=1..3), w=1..3)}; WWlist[m]:={seq(seq(WW[m,r,v], r=1..3), v=1..3)}; for v from 1 by 1 to 3 do VVV[m,v]:={seq(seq(seq(seq(simplify(coeff(RRR[m,v], x[r,t,u,w])), r = 1..3), t = 1..3), u = 1..3), w=1..3)} end do; SSS[m]:=solve(VVV[m,1] union VVV[m,2] union VVV[m,3] union sol_sub_seq[m], {a,b,c,d,e,q,p,f,g,h,j,k} union YYlist[m] union WWlist[m]); TTT1[m]:=subs(gone_sub_list3[m], add(add(add(YY[m,t,u,w]*x[t,u,w], t=1..3), u=1..3), w=1..3)); for v from 1 by 1 to 3 do TTT2[m,v]:=add(WW[m,r,v]*x[r], r =1..3) end do end do; for m from 1 by 1 to nops(sol_sub_seq) do print("_______________________________________________"); print("_______________________________________________"); print("finding normal elements for solution number", m); print("_______________________________________________"); print(sol_sub_seq[m]); print("_______________________________________________"); for u from 1 by 1 to nops({SSS[m]}) do print("deg 3normal element", u); NE3[m,u]:=simplify(subs({SSS[m]}[u], TTT1[m])); if NE3[m,u]<>0 then print("normal element=",NE3[m,u]); for v from 1 by 1 to 3 do print(x[v], "*", NE3[m,u], "=", NE3[m,u], "*", simplify(subs({SSS[m]}[u], TTT2[m,v]))) end do; print("a=", simplify(subs({SSS[m]}[u], a)), "b=", simplify(subs({SSS[m]}[u], b)), "c=", simplify(subs({SSS[m]}[u], c)), "d=", simplify(subs({SSS[m]}[u], d)), "e=", simplify(subs({SSS[m]}[u], e)), "f=", simplify(subs({SSS[m]}[u], f)), "g=", simplify(subs({SSS[m]}[u], g)), "h=", simplify(subs({SSS[m]}[u], h)), "j=", simplify(subs({SSS[m]}[u], j)), "k=", simplify(subs({SSS[m]}[u], k)), "q=", simplify(subs({SSS[m]}[u], q))); print("-------------------------------------------------------------------") end if end do end do; ############################################## #The program above finds three normal elements: #x_1^2x_2, x_3^3, x_1^3 + x_2^3. #We now want to add more relations to the program to #mod out by these normal elements, #and look for normal elements of the factor ring. #It turned out #that one has to look in degree 6 to find the last #normal element, so #below we give the program to look for that degree 6 element. #the relation x_1^5=0 is also a consequence of the others #we already have and so we also add that in. #This program is not quite a generalization of #the earlier ones but is #a one-off version designed just to handle this single case. #The program takes a long time to run. ############################################# rr1d7_sub_list:={seq(seq(seq(seq(seq(x[u,v,r,s,t,2,1]= p*x[p,q,r,s,t,1,2], r = 1..3), s = 1..3), t= 1..3), u=1..3), v=1..3)} union {seq(seq(seq(seq(seq(x[u,v,r,s,2,1,t]= p*x[u,v,r,s,1,2,t], r = 1..3), s = 1..3), t= 1..3), u=1..3), v=1..3)} union {seq(seq(seq(seq(seq(x[u,v,r,2,1,s,t]= p*x[u,v,r,1,2,s,t], r = 1..3), s = 1..3), t= 1..3), u=1..3), v=1..3)} union {seq(seq(seq(seq(seq(x[u,v,2,1,r,s,t]= p*x[u,v,1,2,r,s,t], r = 1..3), s = 1..3), t= 1..3), u=1..3), v=1..3)} union {seq(seq(seq(seq(seq(x[u,2,1,r,s,t,v]= p*x[u,1,2,r,s,t,v], r = 1..3), s = 1..3), t= 1..3), u=1..3), v=1..3)} union {seq(seq(seq(seq(seq(x[2,1,r,s,t,u,v]= p*x[1,2,r,s,t,u,v], r = 1..3), s = 1..3), t= 1..3), u=1..3), v=1..3)}; rr2d7_sub_list:= {seq(seq(seq(seq(seq(x[u,v,r,s,t,3,2]= a*x[u,v,r,s,t,3,1] +x[u,v,r,s,t,1,3]+b*x[u,v,r,s,t,2,3], r = 1..3), s = 1..3), t= 1..3), u=1..3), v=1..3)} union {seq(seq(seq(seq(seq(x[v,r,s,t,3,2,u]= a*x[v,r,s,t,3,1,u] +x[v,r,s,t,1,3,u]+b*x[v,r,s,t,2,3,u], r = 1..3), s = 1..3), t= 1..3), u=1..3), v=1..3)} union {seq(seq(seq(seq(seq(x[r,s,t,3,2,u,v]= a*x[r,s,t,3,1,u,v] +x[r,s,t,1,3,u,v]+b*x[r,s,t,2,3,u,v], r = 1..3), s = 1..3), t= 1..3), u=1..3), v=1..3)} union {seq(seq(seq(seq(seq(x[s,t,3,2,u,v,r]= a*x[s,t,3,1,u,v,r] +x[s,t,1,3,u,v,r]+b*x[s,t,2,3,u,v,r], r = 1..3), s = 1..3), t= 1..3), u=1..3), v=1..3)} union {seq(seq(seq(seq(seq(x[t,3,2,u,v,r,s]= a*x[t,3,1,u,v,r,s] +x[t,1,3,u,v,r,s]+b*x[t,2,3,u,v,r,s], r = 1..3), s = 1..3), t= 1..3), u=1..3), v=1..3)} union {seq(seq(seq(seq(seq(x[3,2,u,v,r,s,t]= a*x[3,1,u,v,r,s,t] +x[1,3,u,v,r,s,t]+b*x[2,3,u,v,r,s,t], r = 1..3), s = 1..3), t= 1..3), u=1..3), v=1..3)}; rr3d7_sub_list:= {seq(seq(seq(seq(x[t,u,r,s,3,3,1]= c*x[t,u,r,s,1,3,3] +d*x[t,u,r,s,2,3,3]+e*x[t,u,r,s,3,1,3], r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[u,r,s,3,3,1,t]= c*x[u,r,s,1,3,3,t] +d*x[u,r,s,2,3,3,t]+e*x[u,r,s,3,1,3,t], r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[r,s,3,3,1,t,u]= c*x[r,s,1,3,3,t,u] +d*x[r,s,2,3,3,t,u]+e*x[r,s,3,1,3,t,u], r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[s,3,3,1,t,u,r]= c*x[s,1,3,3,t,u,r] +d*x[s,2,3,3,t,u,r]+e*x[s,3,1,3,t,u,r], r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[3,3,1,t,u,r,s]= c*x[1,3,3,t,u,r,s] +d*x[2,3,3,t,u,r,s]+e*x[3,1,3,t,u,r,s], r = 1..3), s = 1..3), t=1..3), u=1..3)}; rr4d7_sub_list:= {seq(seq(seq(seq(x[t,u,r,s,3,1,1]= f*x[t,u,r,s,1,1,3] +g*x[t,u,r,s,1,2,3]+h*x[t,u,r,s,2,2,3]+j*x[t,u,r,s,1,3,1]+k*x[t,u,r,s,2,3,1], r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[u,r,s,3,1,1,t]= f*x[u,r,s,1,1,3,t] +g*x[u,r,s,1,2,3,t]+h*x[u,r,s,2,2,3,t]+j*x[u,r,s,1,3,1,t]+k*x[u,r,s,2,3,1,t], r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[r,s,3,1,1,t,u]= f*x[r,s,1,1,3,t,u] +g*x[r,s,1,2,3,t,u]+h*x[r,s,2,2,3,t,u]+j*x[r,s,1,3,1,t,u]+k*x[r,s,2,3,1,t,u], r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[s,3,1,1,t,u,r]= f*x[s,1,1,3,t,u,r] +g*x[s,1,2,3,t,u,r]+h*x[s,2,2,3,t,u,r]+j*x[s,1,3,1,t,u,r]+k*x[s,2,3,1,t,u,r], r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[3,1,1,t,u,r,s]= f*x[1,1,3,t,u,r,s] +g*x[1,2,3,t,u,r,s]+h*x[2,2,3,t,u,r,s]+j*x[1,3,1,t,u,r,s]+k*x[2,3,1,t,u,r,s], r = 1..3), s = 1..3), t=1..3), u=1..3)}; rr5d7_sub_list:= {seq(seq(seq(seq(x[t,u,r,s,3,1,2]= q*x[t,u,r,s,1,3,1] +q*b*x[t,u,r,s,2,3,1]+a*q*x[t,u,r,s,3,1,1], r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[u,r,s,3,1,2,t]= q*x[u,r,s,1,3,1,t] +q*b*x[u,r,s,2,3,1,t]+a*q*x[u,r,s,3,1,1,t], r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[r,s,3,1,2,t,u]= q*x[r,s,1,3,1,t,u] +q*b*x[r,s,2,3,1,t,u]+a*q*x[r,s,3,1,1,t,u], r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[s,3,1,2,t,u,r]= q*x[s,1,3,1,t,u,r] +q*b*x[s,2,3,1,t,u,r]+a*q*x[s,3,1,1,t,u,r], r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[3,1,2,t,u,r,s]= q*x[1,3,1,t,u,r,s] +q*b*x[2,3,1,t,u,r,s]+a*q*x[3,1,1,t,u,r,s], r = 1..3), s = 1..3), t=1..3), u=1..3)}; rr6d7_sub_list:= {seq(seq(seq(seq(x[t,u,r,s,3,3,3]= 0, r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[u,r,s,3,3,3,t]= 0, r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[r,s,3,3,3,t,u]=0, r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[s,3,3,3,t,u,r]=0, r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[3,3,3,t,u,r,s]=0, r = 1..3), s = 1..3), t=1..3), u=1..3)}; rr7d7_sub_list:= {seq(seq(seq(seq(x[t,u,r,s,1,1,2]= 0, r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[u,r,s,1,1,2,t]= 0, r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[r,s,1,1,2,t,u]=0, r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[s,1,1,2,t,u,r]=0, r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[1,1,2,t,u,r,s]=0, r = 1..3), s = 1..3), t=1..3), u=1..3)}; rr8d7_sub_list:= {seq(seq(seq(seq(x[t,u,r,s,2,2,2]= -x[t,u,r,s,1,1,1], r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[u,r,s,2,2,2,t]= -x[u,r,s,1,1,1,t], r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[r,s,2,2,2,t,u]= -x[r,s,1,1,1,t,u], r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[s,2,2,2,t,u,r]= -x[s,1,1,1,t,u,r], r = 1..3), s = 1..3), t=1..3), u=1..3)} union {seq(seq(seq(seq(x[2,2,2,t,u,r,s]= -x[1,1,1,t,u,r,s], r = 1..3), s = 1..3), t=1..3), u=1..3)}; rr9d7_sub_list:= {seq(seq(x[t,u,1,1,1,1,1]= 0, t = 1..3), u = 1..3)} union {seq(seq(x[t,1,1,1,1,1,u]= 0, t = 1..3), u = 1..3)} union {seq(seq(x[1,1,1,1,1,t,u]= 0, t = 1..3), u = 1..3)}; sol_sub_seq:=[{e = -1, h = 0, f = 0, j = 0, c = 0, q = RootOf(_Z^2+_Z+1), p = -1-RootOf(_Z^2+_Z+1), a = -RootOf(_Z^2+_Z+1), d = -1-RootOf(_Z^2+_Z+1), b = 1, g = RootOf(_Z^2+_Z+1), k = -1-RootOf(_Z^2+_Z+1)}]; m:=1; r1d7_sub_list[m]:=subs(sol_sub_seq[m], rr1d7_sub_list); r2d7_sub_list[m]:=subs(sol_sub_seq[m], rr2d7_sub_list); r3d7_sub_list[m]:=subs(sol_sub_seq[m], rr3d7_sub_list); r4d7_sub_list[m]:=subs(sol_sub_seq[m], rr4d7_sub_list); r5d7_sub_list[m]:=subs(sol_sub_seq[m], rr5d7_sub_list); r6d7_sub_list[m]:=subs(sol_sub_seq[m], rr6d7_sub_list); r7d7_sub_list[m]:=subs(sol_sub_seq[m], rr7d7_sub_list); r8d7_sub_list[m]:=subs(sol_sub_seq[m], rr8d7_sub_list); r9d7_sub_list[m]:=subs(sol_sub_seq[m], rr9d7_sub_list); NE:= Y[3,1,3,1,3,1]*x[3,1,3,1,3,1] +Y[3,1,3,1,3,3]*x[3,1,3,1,3,3] +Y[1,3,1,3,1,3]*x[1,3,1,3,1,3] +Y[2,3,1,3,1,3]*x[2,3,1,3,1,3] +Y[1,1,3,1,3,1]*x[1,1,3,1,3,1] +Y[1,2,3,1,3,1]*x[1,2,3,1,3,1] +Y[2,2,3,1,3,1]*x[2,2,3,1,3,1] +Y[1,1,1,1,3,1]*x[1,1,1,1,3,1] +Y[1,1,1,3,1,3]*x[1,1,1,3,1,3] +Y[1,2,2,3,1,3]*x[1,2,2,3,1,3] +Y[1,1,3,1,3,3]*x[1,1,3,1,3,3] +Y[1,2,3,1,3,3]*x[1,2,3,1,3,3] +Y[2,2,3,1,3,3]*x[2,2,3,1,3,3] +Y[1,1,1,1,3,3]*x[1,1,1,1,3,3]; for r from 1 by 1 to 3 do NEtimes[r]:= Y[3,1,3,1,3,1]*x[3,1,3,1,3,1,r] +Y[3,1,3,1,3,3]*x[3,1,3,1,3,3,r] +Y[1,3,1,3,1,3]*x[1,3,1,3,1,3,r] +Y[2,3,1,3,1,3]*x[2,3,1,3,1,3,r] +Y[1,1,3,1,3,1]*x[1,1,3,1,3,1,r] +Y[1,2,3,1,3,1]*x[1,2,3,1,3,1,r] +Y[2,2,3,1,3,1]*x[2,2,3,1,3,1,r] +Y[1,1,1,1,3,1]*x[1,1,1,1,3,1,r] +Y[1,1,1,3,1,3]*x[1,1,1,3,1,3,r] +Y[1,2,2,3,1,3]*x[1,2,2,3,1,3,r] +Y[1,1,3,1,3,3]*x[1,1,3,1,3,3,r] +Y[1,2,3,1,3,3]*x[1,2,3,1,3,3,r] +Y[2,2,3,1,3,3]*x[2,2,3,1,3,3,r] +Y[1,1,1,1,3,3]*x[1,1,1,1,3,3,r] end do; for r from 1 by 1 to 3 do Otherside[r]:=add(A[r,s]*Y[3,1,3,1,3,1]*x[s,3,1,3,1,3,1] +A[r,s]*Y[3,1,3,1,3,3]*x[s,3,1,3,1,3,3] +A[r,s]*Y[1,3,1,3,1,3]*x[s,1,3,1,3,1,3] +A[r,s]*Y[2,3,1,3,1,3]*x[s,2,3,1,3,1,3] +A[r,s]*Y[1,1,3,1,3,1]*x[s,1,1,3,1,3,1] +A[r,s]*Y[1,2,3,1,3,1]*x[s,1,2,3,1,3,1] +A[r,s]*Y[2,2,3,1,3,1]*x[s,2,2,3,1,3,1] +A[r,s]*Y[1,1,1,1,3,1]*x[s,1,1,1,1,3,1] +A[r,s]*Y[1,1,1,3,1,3]*x[s,1,1,1,3,1,3] +A[r,s]*Y[1,2,2,3,1,3]*x[s,1,2,2,3,1,3] +A[r,s]*Y[1,1,3,1,3,3]*x[s,1,1,3,1,3,3] +A[r,s]*Y[1,2,3,1,3,3]*x[s,1,2,3,1,3,3] +A[r,s]*Y[2,2,3,1,3,3]*x[s,2,2,3,1,3,3] +A[r,s]*Y[1,1,1,1,3,3]*x[s,1,1,1,1,3,3], s=1..3) end do; for r from 1 by 1 to 3 do RR1[r]:=NEtimes[r] - Otherside[r] end do; for r from 1 by 1 to 3 do RR2[r]:=subs(r9d7_sub_list[m],subs(r8d7_sub_list[m],subs(r7d7_sub_list[m],subs(r6d7_sub_list[m], subs(r5d7_sub_list[m], subs(r4d7_sub_list[m], subs(r3d7_sub_list[m], subs(r2d7_sub_list[m],subs(r1d7_sub_list[m], RR1[r]))))))))) end do; for r from 1 by 1 to 3 do RR3[r]:=subs(r9d7_sub_list[m],subs(r8d7_sub_list[m],subs(r7d7_sub_list[m],subs(r6d7_sub_list[m], subs(r5d7_sub_list[m], subs(r4d7_sub_list[m], subs(r3d7_sub_list[m], subs(r2d7_sub_list[m],subs(r1d7_sub_list[m], RR2[r]))))))))) end do; for r from 1 by 1 to 3 do RR4[r]:=subs(r9d7_sub_list[m],subs(r8d7_sub_list[m],subs(r7d7_sub_list[m],subs(r6d7_sub_list[m], subs(r5d7_sub_list[m], subs(r4d7_sub_list[m], subs(r3d7_sub_list[m], subs(r2d7_sub_list[m],subs(r1d7_sub_list[m], RR3[r]))))))))) end do; for r from 1 by 1 to 3 do RR5[r]:=subs(r9d7_sub_list[m],subs(r8d7_sub_list[m],subs(r7d7_sub_list[m],subs(r6d7_sub_list[m], subs(r5d7_sub_list[m], subs(r4d7_sub_list[m], subs(r3d7_sub_list[m], subs(r2d7_sub_list[m],subs(r1d7_sub_list[m], RR4[r]))))))))) end do; for r from 1 by 1 to 3 do RR6[r]:=subs(r9d7_sub_list[m],subs(r8d7_sub_list[m],subs(r7d7_sub_list[m],subs(r6d7_sub_list[m], subs(r5d7_sub_list[m], subs(r4d7_sub_list[m], subs(r3d7_sub_list[m], subs(r2d7_sub_list[m],subs(r1d7_sub_list[m], RR5[r]))))))))) end do; for r from 1 by 1 to 3 do RR7[r]:=subs(r9d7_sub_list[m],subs(r8d7_sub_list[m],subs(r7d7_sub_list[m],subs(r6d7_sub_list[m], subs(r5d7_sub_list[m], subs(r4d7_sub_list[m], subs(r3d7_sub_list[m], subs(r2d7_sub_list[m],subs(r1d7_sub_list[m], RR6[r]))))))))) end do; for r from 1 by 1 to 3 do RR8[r]:=subs(r9d7_sub_list[m],subs(r8d7_sub_list[m],subs(r7d7_sub_list[m],subs(r6d7_sub_list[m], subs(r5d7_sub_list[m], subs(r4d7_sub_list[m], subs(r3d7_sub_list[m], subs(r2d7_sub_list[m],subs(r1d7_sub_list[m], RR7[r]))))))))) end do; for r from 1 by 1 to 3 do RR[r]:=subs(r9d7_sub_list[m],subs(r8d7_sub_list[m],subs(r7d7_sub_list[m],subs(r6d7_sub_list[m], subs(r5d7_sub_list[m], subs(r4d7_sub_list[m], subs(r3d7_sub_list[m], subs(r2d7_sub_list[m],subs(r1d7_sub_list[m], RR8[r]))))))))) end do; mon_list:={x[1,3,1,3,1,3,1],x[2,3,1,3,1,3,1] ,x[3,1,3,1,3,1,3] ,x[1,3,1,3,1,3,3] ,x[2,3,1,3,1,3,3] ,x[1,1,3,1,3,1,3] ,x[1,2,3,1,3,1,3] ,x[2,2,3,1,3,1,3] ,x[1,2,2,3,1,3,1] ,x[1,1,1,3,1,3,1] ,x[1,1,1,3,1,3,3] ,x[1,2,2,3,1,3,3] ,x[1,1,1,1,3,1,3]}; for r from 1 by 1 to 3 do for v from 1 by 1 to nops(mon_list) do equats[v,r]:=simplify(coeff(RR[r], mon_list[v])) end do end do; solvelist:={seq(seq(equats[v,r], r=1..3), v=1..nops(mon_list))}; Ylist:={Y[3,1,3,1,3,1] ,Y[3,1,3,1,3,3] ,Y[1,3,1,3,1,3] ,Y[2,3,1,3,1,3] ,Y[1,1,3,1,3,1] ,Y[1,2,3,1,3,1] ,Y[2,2,3,1,3,1] ,Y[1,1,1,1,3,1] ,Y[1,1,1,3,1,3] ,Y[1,2,2,3,1,3] ,Y[1,1,3,1,3,3] ,Y[1,2,3,1,3,3] ,Y[2,2,3,1,3,3] ,Y[1,1,1,1,3,3]}; Alist:={seq(seq(A[r,s], r=1..3), s=1..3)}; S:=solve(solvelist, Ylist union Alist); for s from 1 by 1 to nops([S]) do print("-------------------------------------------------------------------"); print("deg", 6, "normal element", s); print("normal element N=",subs(S[s], NE)); for r from 1 by 1 to 3 do print(" "); print(subs(S[s], A[r,1]*x[1] + A[r,2]*x[2]+ A[r,3]*x[3]),"*", N , "=", N, "*", x[r]) end do end do; ##################### #The program finds (among others) the #degree 6 normal element claimed in the paper; #in fact the program claims the element is central. #Since the program is so complicated, below we offer #a final program verifying that the desired element N is indeed #central. Here, R[i] = N*x_i and L[i] = x_i*N. ##################### R[1]:=x[3,1,3,1,3,1,1] + x[1,3,1,3,1,3,1] +(RootOf(_Z^2+_Z+1) + 1)*x[2,3,1,3,1,3,1] - x[1,1,3,1,3,3,1] - x[1,2,3,1,3,3,1]; L[1]:=x[1,3,1,3,1,3,1] + x[1,1,3,1,3,1,3] +(RootOf(_Z^2+_Z+1) + 1)*x[1,2,3,1,3,1,3] - x[1,1,1,3,1,3,3] - x[1,1,2,3,1,3,3]; R[2]:=x[3,1,3,1,3,1,2] + x[1,3,1,3,1,3,2] +(RootOf(_Z^2+_Z+1) + 1)*x[2,3,1,3,1,3,2] - x[1,1,3,1,3,3,2] - x[1,2,3,1,3,3,2]; L[2]:=x[2,3,1,3,1,3,1] + x[2,1,3,1,3,1,3] +(RootOf(_Z^2+_Z+1) + 1)*x[2,2,3,1,3,1,3] - x[2,1,1,3,1,3,3] - x[2,1,2,3,1,3,3]; R[3]:=x[3,1,3,1,3,1,3] + x[1,3,1,3,1,3,3] +(RootOf(_Z^2+_Z+1) + 1)*x[2,3,1,3,1,3,3] - x[1,1,3,1,3,3,3] - x[1,2,3,1,3,3,3]; L[3]:=x[3,3,1,3,1,3,1] + x[3,1,3,1,3,1,3] +(RootOf(_Z^2+_Z+1) + 1)*x[3,2,3,1,3,1,3] - x[3,1,1,3,1,3,3] - x[3,1,2,3,1,3,3]; for r from 1 by 1 to 3 do RR1[r]:=R[r]-L[r] end do; for r from 1 by 1 to 3 do RR2[r]:=subs(r9d7_sub_list[m],subs(r8d7_sub_list[m],subs(r7d7_sub_list[m],subs(r6d7_sub_list[m], subs(r5d7_sub_list[m], subs(r4d7_sub_list[m], subs(r3d7_sub_list[m], subs(r2d7_sub_list[m],subs(r1d7_sub_list[m], RR1[r]))))))))) end do; for r from 1 by 1 to 3 do RR3[r]:=subs(r9d7_sub_list[m],subs(r8d7_sub_list[m],subs(r7d7_sub_list[m],subs(r6d7_sub_list[m], subs(r5d7_sub_list[m], subs(r4d7_sub_list[m], subs(r3d7_sub_list[m], subs(r2d7_sub_list[m],subs(r1d7_sub_list[m], RR2[r]))))))))) end do; for r from 1 by 1 to 3 do RR4[r]:=subs(r9d7_sub_list[m],subs(r8d7_sub_list[m],subs(r7d7_sub_list[m],subs(r6d7_sub_list[m], subs(r5d7_sub_list[m], subs(r4d7_sub_list[m], subs(r3d7_sub_list[m], subs(r2d7_sub_list[m],subs(r1d7_sub_list[m], RR3[r]))))))))) end do; for r from 1 by 1 to 3 do RR5[r]:=subs(r9d7_sub_list[m],subs(r8d7_sub_list[m],subs(r7d7_sub_list[m],subs(r6d7_sub_list[m], subs(r5d7_sub_list[m], subs(r4d7_sub_list[m], subs(r3d7_sub_list[m], subs(r2d7_sub_list[m],subs(r1d7_sub_list[m], RR4[r]))))))))) end do; for r from 1 by 1 to 3 do RR6[r]:=subs(r9d7_sub_list[m],subs(r8d7_sub_list[m],subs(r7d7_sub_list[m],subs(r6d7_sub_list[m], subs(r5d7_sub_list[m], subs(r4d7_sub_list[m], subs(r3d7_sub_list[m], subs(r2d7_sub_list[m],subs(r1d7_sub_list[m], RR5[r]))))))))) end do; for r from 1 by 1 to 3 do RR7[r]:=subs(r9d7_sub_list[m],subs(r8d7_sub_list[m],subs(r7d7_sub_list[m],subs(r6d7_sub_list[m], subs(r5d7_sub_list[m], subs(r4d7_sub_list[m], subs(r3d7_sub_list[m], subs(r2d7_sub_list[m],subs(r1d7_sub_list[m], RR6[r]))))))))) end do; for r from 1 by 1 to 3 do RR8[r]:=subs(r9d7_sub_list[m],subs(r8d7_sub_list[m],subs(r7d7_sub_list[m],subs(r6d7_sub_list[m], subs(r5d7_sub_list[m], subs(r4d7_sub_list[m], subs(r3d7_sub_list[m], subs(r2d7_sub_list[m],subs(r1d7_sub_list[m], RR7[r]))))))))) end do; for r from 1 by 1 to 3 do RR[r]:=subs(r9d7_sub_list[m],subs(r8d7_sub_list[m],subs(r7d7_sub_list[m],subs(r6d7_sub_list[m], subs(r5d7_sub_list[m], subs(r4d7_sub_list[m], subs(r3d7_sub_list[m], subs(r2d7_sub_list[m],subs(r1d7_sub_list[m], RR8[r]))))))))) end do; for r from 1 by 1 to 3 do print(simplify(RR[r])) end do; #################### #The output from this program is three zeroes, as desired. ###################### #################### #Below, we reproduce the output from the first two programs #above. #################### "_______________________________________________" "_______________________________________________" "finding normal elements for solution number", 1 "_______________________________________________" {a = -%1, b = 1, c = 0, d = -1 - %1, e = -1, f = 0, g = %1, h = 0, j = 0, k = -1 - %1, p = -1 - %1, q = %1} 2 %1 := RootOf(_Z + _Z + 1) "_______________________________________________" "deg 3normal element", 1 "normal element=", YY[1, 1, 1, 1] x[2, 2, 2] + YY[1, 1, 1, 1] x[1, 1, 1] x[1], "*", YY[1, 1, 1, 1] x[2, 2, 2] + YY[1, 1, 1, 1] x[1, 1, 1], "=", YY[1, 1, 1, 1] x[2, 2, 2] + YY[1, 1, 1, 1] x[1, 1, 1], "*", x[1] x[2], "*", YY[1, 1, 1, 1] x[2, 2, 2] + YY[1, 1, 1, 1] x[1, 1, 1], "=", YY[1, 1, 1, 1] x[2, 2, 2] + YY[1, 1, 1, 1] x[1, 1, 1], "*", x[2] x[3], "*", YY[1, 1, 1, 1] x[2, 2, 2] + YY[1, 1, 1, 1] x[1, 1, 1], "=", YY[1, 1, 1, 1] x[2, 2, 2] + YY[1, 1, 1, 1] x[1, 1, 1], "*", x[3] "a=", -%1, "b=", 1, "c=", 0, "d=", -1 - %1, "e=", -1, "f=", 0, "g=", %1, "h=", 0, "j=", 0, "k=", -1 - %1, "q=", %1 2 %1 := RootOf(_Z + _Z + 1) "---------------------------------------------------------------\ ----" "deg 3normal element", 2 "deg 3normal element", 3 "normal element=", YY[1, 3, 3, 3] x[3, 3, 3] x[1], "*", YY[1, 3, 3, 3] x[3, 3, 3], "=", 2 YY[1, 3, 3, 3] x[3, 3, 3], "*", x[1] RootOf(_Z + _Z + 1) x[2], "*", YY[1, 3, 3, 3] x[3, 3, 3], "=", 2 YY[1, 3, 3, 3] x[3, 3, 3], "*", RootOf(_Z + _Z + 1) x[2] x[3], "*", YY[1, 3, 3, 3] x[3, 3, 3], "=", YY[1, 3, 3, 3] x[3, 3, 3], "*", x[3] "a=", -%1, "b=", 1, "c=", 0, "d=", -1 - %1, "e=", -1, "f=", 0, "g=", %1, "h=", 0, "j=", 0, "k=", -1 - %1, "q=", %1 2 %1 := RootOf(_Z + _Z + 1) "---------------------------------------------------------------\ ----" "deg 3normal element", 4 "normal element=", YY[1, 1, 1, 2] x[1, 1, 2] + YY[1, 3, 3, 3] x[3, 3, 3] x[1], "*", YY[1, 1, 1, 2] x[1, 1, 2] + YY[1, 3, 3, 3] x[3, 3, 3], "=", YY[1, 1, 1, 2] x[1, 1, 2] + YY[1, 3, 3, 3] x[3, 3, 3], 2 "*", x[1] RootOf(_Z + _Z + 1) x[2], "*", YY[1, 1, 1, 2] x[1, 1, 2] + YY[1, 3, 3, 3] x[3, 3, 3], "=", YY[1, 1, 1, 2] x[1, 1, 2] + YY[1, 3, 3, 3] x[3, 3, 3], 2 "*", RootOf(_Z + _Z + 1) x[2] x[3], "*", YY[1, 1, 1, 2] x[1, 1, 2] + YY[1, 3, 3, 3] x[3, 3, 3], "=", YY[1, 1, 1, 2] x[1, 1, 2] + YY[1, 3, 3, 3] x[3, 3, 3], "*", x[3] "a=", -%1, "b=", 1, "c=", 0, "d=", -1 - %1, "e=", -1, "f=", 0, "g=", %1, "h=", 0, "j=", 0, "k=", -1 - %1, "q=", %1 2 %1 := RootOf(_Z + _Z + 1) "---------------------------------------------------------------\ ----" ############################################################# "---------------------------------------------------------------\ ----" "deg", 6, "normal element", 1 "normal element N=", -%1 x[1, 1, 3, 1, 3, 1] + %1 x[1, 2, 3, 1, 3, 1] - %1 %2 x[2, 2, 3, 1, 3, 1] - A[3, 1] A[1, 1] %1 x[1, 1, 1, 1, 3, 1] + (-A[1, 1] %1 + %1) x[1, 1, 1, 3, 1, 3] + (-%1 %2 - %1 %2 A[1, 1] - %1) x[1, 2, 2, 3, 1, 3] 2 + (%1 %2 + A[1, 1] %1 - A[1, 1] %1) x[1, 1, 1, 1, 3, 3] %1 := Y[1, 2, 3, 1, 3, 1] 2 %2 := RootOf(_Z + _Z + 1) " " A[1, 1] x[1], "*", N, "=", N, "*", x[1] " " A[1, 1] x[2], "*", N, "=", N, "*", x[2] " " x[3] A[3, 1] x[1] + -------, "*", N, "=", N, "*", x[3] A[1, 1] "---------------------------------------------------------------\ ----" "deg", 6, "normal element", 2 "normal element N=", (-%1 - %2 %1) x[1, 1, 3, 1, 3, 1] + (%1 + %2 %1) x[1, 2, 3, 1, 3, 1] + %1 x[2, 2, 3, 1, 3, 1] + (-A[3, 1] %1 - A[3, 1] %2 %1) x[1, 1, 1, 1, 3, 1] + (%1 - %2 %1) x[1, 2, 2, 3, 1, 3] - %1 x[1, 1, 1, 1, 3, 3] %1 := Y[2, 2, 3, 1, 3, 1] 2 %2 := RootOf(_Z + _Z + 1) " " x[1], "*", N, "=", N, "*", x[1] " " x[2], "*", N, "=", N, "*", x[2] " " A[3, 1] x[1] + x[3], "*", N, "=", N, "*", x[3] "---------------------------------------------------------------\ ----" "deg", 6, "normal element", 3 "normal element N=", %3 x[3, 1, 3, 1, 3, 1] + %3 x[1, 3, 1, 3, 1, 3] + (%3 + %1 %3) x[2, 3, 1, 3, 1, 3] - %2 x[1, 1, 3, 1, 3, 1] + %2 x[1, 2, 3, 1, 3, 1] - %2 %1 x[2, 2, 3, 1, 3, 1] + (-2 %2 %1 - %2) x[1, 2, 2, 3, 1, 3] - %3 x[1, 1, 3, 1, 3, 3] - %3 x[1, 2, 3, 1, 3, 3] + %2 %1 x[1, 1, 1, 1, 3, 3] 2 %1 := RootOf(_Z + _Z + 1) %2 := Y[1, 2, 3, 1, 3, 1] %3 := Y[3, 1, 3, 1, 3, 1] " " x[1], "*", N, "=", N, "*", x[1] " " x[2], "*", N, "=", N, "*", x[2] " " x[3], "*", N, "=", N, "*", x[3] "---------------------------------------------------------------\ ----" "deg", 6, "normal element", 4 "normal element N=", -%2 x[1, 1, 3, 1, 3, 1] + %2 x[1, 2, 3, 1, 3, 1] - %2 %1 x[2, 2, 3, 1, 3, 1] A[3, 1] %2 x[1, 1, 1, 1, 3, 1] - ------------------------------ A[3, 3] %2 (A[3, 3] - 1) x[1, 1, 1, 3, 1, 3] + ------------------------------------ A[3, 3] %2 (%1 + %1 A[3, 3] + A[3, 3]) x[1, 2, 2, 3, 1, 3] - -------------------------------------------------- A[3, 3] 2 %2 (A[3, 3] %1 - A[3, 3] + 1) x[1, 1, 1, 1, 3, 3] + -------------------------------------------------- 2 A[3, 3] 2 %1 := RootOf(_Z + _Z + 1) %2 := Y[1, 2, 3, 1, 3, 1] " " x[1] -------, "*", N, "=", N, "*", x[1] A[3, 3] " " x[2] -------, "*", N, "=", N, "*", x[2] A[3, 3] " " A[3, 1] x[1] + A[3, 3] x[3], "*", N, "=", N, "*", x[3] "---------------------------------------------------------------\ ----" "deg", 6, "normal element", 5 "normal element N=", -%1 x[1, 1, 3, 1, 3, 1] + %1 x[1, 2, 3, 1, 3, 1] - %1 %2 x[2, 2, 3, 1, 3, 1] + Y[1, 1, 1, 1, 3, 1] x[1, 1, 1, 1, 3, 1] + (2 %1 + %1 %2) x[1, 1, 1, 3, 1, 3] + (-%1 %2 - 2 %1) x[1, 2, 2, 3, 1, 3] + (3 %1 %2 + %1) x[1, 1, 1, 1, 3, 3] %1 := Y[1, 2, 3, 1, 3, 1] 2 %2 := RootOf(_Z + _Z + 1) " " 2 (-1 - RootOf(_Z + _Z + 1)) x[1], "*", N, "=", N, "*", x[1] " " 2 (-1 - RootOf(_Z + _Z + 1)) x[2], "*", N, "=", N, "*", x[2] " " 2 Y[1, 1, 1, 1, 3, 1] RootOf(_Z + _Z + 1) x[1] - --------------------------------------------- Y[1, 2, 3, 1, 3, 1] 2 + RootOf(_Z + _Z + 1) x[3], "*", N, "=", N, "*", x[3] "---------------------------------------------------------------\ ----" "deg", 6, "normal element", 6 "normal element N=", -%2 x[1, 1, 3, 1, 3, 1] + %2 x[1, 2, 3, 1, 3, 1] - %2 %1 x[2, 2, 3, 1, 3, 1] A[3, 1] %2 x[1, 1, 1, 1, 3, 1] - ------------------------------ A[3, 3] %2 (A[3, 3] - 1) x[1, 1, 1, 3, 1, 3] + ------------------------------------ A[3, 3] %2 (%1 + %1 A[3, 3] + A[3, 3]) x[1, 2, 2, 3, 1, 3] - -------------------------------------------------- A[3, 3] 2 %2 (A[3, 3] %1 - A[3, 3] + 1) x[1, 1, 1, 1, 3, 3] + -------------------------------------------------- 2 A[3, 3] 2 %1 := RootOf(_Z + _Z + 1) %2 := Y[1, 2, 3, 1, 3, 1] " " x[1] -------, "*", N, "=", N, "*", x[1] A[3, 3] " " x[2] -------, "*", N, "=", N, "*", x[2] A[3, 3] " " A[3, 1] x[1] + A[3, 3] x[3], "*", N, "=", N, "*", x[3] "---------------------------------------------------------------\ ----" "deg", 6, "normal element", 7 "normal element N=", -%2 x[1, 1, 3, 1, 3, 1] + %2 x[1, 2, 3, 1, 3, 1] - %2 %1 x[2, 2, 3, 1, 3, 1] A[3, 1] %2 x[1, 1, 1, 1, 3, 1] - ------------------------------ A[3, 3] %2 (A[3, 3] - 1) x[1, 1, 1, 3, 1, 3] + ------------------------------------ A[3, 3] %2 (%1 + %1 A[3, 3] + A[3, 3]) x[1, 2, 2, 3, 1, 3] - -------------------------------------------------- A[3, 3] 2 %2 (A[3, 3] %1 - A[3, 3] + 1) x[1, 1, 1, 1, 3, 3] + -------------------------------------------------- 2 A[3, 3] 2 %1 := RootOf(_Z + _Z + 1) %2 := Y[1, 2, 3, 1, 3, 1] " " x[1] -------, "*", N, "=", N, "*", x[1] A[3, 3] " " x[2] -------, "*", N, "=", N, "*", x[2] A[3, 3] " " A[3, 1] x[1] + A[3, 3] x[3], "*", N, "=", N, "*", x[3] "---------------------------------------------------------------\ ----" "deg", 6, "normal element", 8 "normal element N=", -%2 x[1, 1, 3, 1, 3, 1] + %2 x[1, 2, 3, 1, 3, 1] - %2 %1 x[2, 2, 3, 1, 3, 1] A[3, 1] %2 x[1, 1, 1, 1, 3, 1] - ------------------------------ A[3, 3] %2 (A[3, 3] - 1) x[1, 1, 1, 3, 1, 3] + ------------------------------------ A[3, 3] %2 (%1 + %1 A[3, 3] + A[3, 3]) x[1, 2, 2, 3, 1, 3] - -------------------------------------------------- A[3, 3] 2 %2 (A[3, 3] %1 - A[3, 3] + 1) x[1, 1, 1, 1, 3, 3] + -------------------------------------------------- 2 A[3, 3] 2 %1 := RootOf(_Z + _Z + 1) %2 := Y[1, 2, 3, 1, 3, 1] " " x[1] -------, "*", N, "=", N, "*", x[1] A[3, 3] " " x[2] -------, "*", N, "=", N, "*", x[2] A[3, 3] " " A[3, 1] x[1] + A[3, 3] x[3], "*", N, "=", N, "*", x[3] "---------------------------------------------------------------\ ----" "deg", 6, "normal element", 9 "normal element N=", -%2 x[1, 1, 3, 1, 3, 1] + %2 x[1, 2, 3, 1, 3, 1] - %2 %1 x[2, 2, 3, 1, 3, 1] A[3, 1] %2 x[1, 1, 1, 1, 3, 1] - ------------------------------ A[3, 3] %2 (A[3, 3] - 1) x[1, 1, 1, 3, 1, 3] + ------------------------------------ A[3, 3] %2 (%1 + %1 A[3, 3] + A[3, 3]) x[1, 2, 2, 3, 1, 3] - -------------------------------------------------- A[3, 3] 2 %2 (A[3, 3] %1 - A[3, 3] + 1) x[1, 1, 1, 1, 3, 3] + -------------------------------------------------- 2 A[3, 3] 2 %1 := RootOf(_Z + _Z + 1) %2 := Y[1, 2, 3, 1, 3, 1] " " x[1] -------, "*", N, "=", N, "*", x[1] A[3, 3] " " x[2] -------, "*", N, "=", N, "*", x[2] A[3, 3] " " A[3, 1] x[1] + A[3, 3] x[3], "*", N, "=", N, "*", x[3] "---------------------------------------------------------------\ ----" "deg", 6, "normal element", 10 "normal element N=", -%2 x[1, 1, 3, 1, 3, 1] + %2 x[1, 2, 3, 1, 3, 1] - %2 %1 x[2, 2, 3, 1, 3, 1] A[3, 1] %2 x[1, 1, 1, 1, 3, 1] - ------------------------------ A[3, 3] %2 (A[3, 3] - 1) x[1, 1, 1, 3, 1, 3] + ------------------------------------ A[3, 3] %2 (%1 + %1 A[3, 3] + A[3, 3]) x[1, 2, 2, 3, 1, 3] - -------------------------------------------------- A[3, 3] 2 %2 (A[3, 3] %1 - A[3, 3] + 1) x[1, 1, 1, 1, 3, 3] + -------------------------------------------------- 2 A[3, 3] 2 %1 := RootOf(_Z + _Z + 1) %2 := Y[1, 2, 3, 1, 3, 1] " " x[1] -------, "*", N, "=", N, "*", x[1] A[3, 3] " " x[2] -------, "*", N, "=", N, "*", x[2] A[3, 3] " " A[3, 1] x[1] + A[3, 3] x[3], "*", N, "=", N, "*", x[3] "---------------------------------------------------------------\ ----" "deg", 6, "normal element", 11 "normal element N=", 0 " " A[1, 1] x[1] + A[1, 2] x[2] + A[1, 3] x[3], "*", N, "=", N, "*", x[1] " " A[2, 1] x[1] + A[2, 2] x[2] + A[2, 3] x[3], "*", N, "=", N, "*", x[2] " " A[3, 1] x[1] + A[3, 2] x[2] + A[3, 3] x[3], "*", N, "=", N, "*", x[3] >