
theorem 2split:
FAdj(F_Rat,{2-Root(2)}) is SplittingField of X^2-2
proof
set F = FAdj(F_Rat,{2-Root(2)});
C: X-2-Root(2) = rpoly(1,2-Root(2)) & X+2-Root(2) = rpoly(1,-2-Root(2))
      by FIELD_9:def 2,FIELD_9:def 3;
   rpoly(1,2-Root(2)) is Ppoly of F_Real &
   rpoly(1,-2-Root(2)) is Ppoly of F_Real by RING_5:51; then
A: rpoly(1,2-Root(2)) *' rpoly(1,-2-Root(2)) is Ppoly of F_Real by RING_5:52;
   X^2-2 = 1.F_Real * (rpoly(1,2-Root(2)) *' rpoly(1,-2-Root(2)))
      by C,POLYNOM3:def 10,2splita; then
D: X^2-2 splits_in F_Real by A,FIELD_4:def 5;
   {2-Root(2), -2-Root(2)} c= the carrier of F
     proof
     2-Root(2) in {2-Root(2)} & {2-Root(2)} is Subset of F
            by FIELD_6:35,TARSKI:def 1; then
     reconsider a = 2-Root(2) as Element of F;
     H: F is Subring of F_Real by FIELD_5:12;
     now let o be object;
       assume o in {2-Root(2), -2-Root(2)}; then
       per cases by TARSKI:def 2;
       suppose o = 2-Root(2);
         then o = a;
         hence o in the carrier of F;
         end;
       suppose o = -2-Root(2);
         then o = -a by H,FIELD_6:17;
         hence o in the carrier of F;
         end;
       end;
     hence thesis;
     end; then
B: X^2-2 splits_in F by D,2splitb,FIELD_8:27;
now let E be FieldExtension of F_Rat;
  assume C: X^2-2 splits_in E & E is Subfield of F; then
  E: F is E-extending & E is Subfield of F_Real by EC_PF_1:5,FIELD_4:7;
  F: F_Rat is Subfield of E by FIELD_4:7;
  {2-Root(2)} is Subset of E
    proof
    G: Roots(F_Real,X^2-2) c= the carrier of E by E,D,C,FIELD_8:27;
    2-Root(2) in Roots(F_Real,X^2-2) by 2splitb,TARSKI:def 2; then
    2-Root(2) in the carrier of E by G;
    then {2-Root(2)} c= the carrier of E by TARSKI:def 1;
    hence thesis;
    end;
  then F is Subfield of E by F,E,FIELD_6:37;
  hence E == F by C,FIELD_7:def 2;
  end;
hence F is SplittingField of X^2-2 by B,FIELD_8:def 1;
end;
