
theorem lemZ2roots:
Roots(FAdj(Z/2,{alpha}),X^2+X+1) = { alpha, alpha-1 }
proof
set F = FAdj(Z/2,{alpha});
H: embField(canHomP X^2+X+1) is FieldExtension of F by FIELD_4:7;
alpha in {alpha} & {alpha} is Subset of F by TARSKI:def 1,FIELD_6:35; then
reconsider a = alpha as Element of F;
alpha" = a" by FIELD_6:18; then
reconsider ai =  alpha" as Element of F;
C: rpoly(1,alpha) = rpoly(1,a) & rpoly(1,alpha") = rpoly(1,ai) by H,FIELD_4:21;
reconsider q = rpoly(1,a) *' rpoly(1,ai)
   as Element of the carrier of Polynom-Ring F by POLYNOM3:def 10;
A: X^2+X+1 = rpoly(1,a) *' rpoly(1,ai) by H,C,lemZ2,FIELD_4:17;
Roots rpoly(1,a) = {a} & Roots rpoly(1,ai) = {ai} by RING_5:18; then
Roots(rpoly(1,a) *' rpoly(1,ai))
   = {a} \/ {ai} by UPROOTS:23
  .= {a,ai} by ENUMSET1:1; then
{a,ai} = Roots q .= Roots(F,X^2+X+1) by A,FIELD_7:13;
hence thesis by lemalph;
end;
