
theorem
FAdj(Z/2,{alpha}) is SplittingField of X^2+X+1
proof
set p = X^2+X+1, 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;
B: X^2+X+1 = 1.F * (rpoly(1,a) *' rpoly(1,ai)) by lemZ2,H,C,FIELD_4:17;
rpoly(1,a) is Ppoly of F & rpoly(1,ai) is Ppoly of F by RING_5:51; then
rpoly(1,a) *' rpoly(1,ai) is Ppoly of F by RING_5:52; then
A: p splits_in F by B,FIELD_4:def 5;
now let E be FieldExtension of Z/2;
  assume D0: p splits_in E & E is Subfield of F; then
  D3: F is E-extending by FIELD_4:7;
  D4: Roots(F,p) c= the carrier of E by D3,D0,A,FIELD_8:27;
  alpha in Roots(F,p) by lemZ2roots,TARSKI:def 2; then
  alpha in the carrier of E by D4; then
  D1: {alpha} c= the carrier of E by TARSKI:def 1;
  D2: E is Subfield of embField(canHomP X^2+X+1) by D0,EC_PF_1:5;
  Z/2 is Subfield of E by FIELD_4:7;
  then F is Subfield of E by D1,D2,FIELD_6:37;
  hence E == F by D0,FIELD_7:def 2;
  end;
hence thesis by A,FIELD_8:def 1;
end;
