reserve
F for non 2-characteristic non quadratic_complete polynomial_disjoint Field;
reserve
p for non DC-square quadratic Element of the carrier of Polynom-Ring F;

theorem
FAdj(F,{sqrt(DC p)}) is SplittingField of p
proof
set E = FAdj(F,{sqrt(DC p)}), r1 = Root1 p, r2 = Root2 p;
reconsider q = rpoly(1,r1) *' rpoly(1,r2) as
                        Element of the carrier of Polynom-Ring E
   by POLYNOM3:def 10;
K: F is Subring of E by FIELD_4:def 1;
Y: now assume Y1: @(LC p,E) is zero;
     LC p = @(LC p,E) by FIELD_7:def 4 .= 0.F by Y1,K,C0SP1:def 3;
     hence contradiction;
     end;
   rpoly(1,r1) *' rpoly(1,r2) = (X-r1) *' (X-r2); then
D: p = @(LC p,FAdj(F,{sqrt(DC p)})) * (rpoly(1,r1) *' rpoly(1,r2)) by Z5;
   rpoly(1,r1) is Ppoly of E & rpoly(1,r2) is Ppoly of E by RING_5:51;
   then rpoly(1,r1) *' rpoly(1,r2) is Ppoly of E by RING_5:52; then
A: p splits_in E by Y,D,FIELD_4:def 5;
now let U be FieldExtension of F;
   assume D0: p splits_in U & U is Subfield of E; then
   D3: E is U-extending & U is Subring of E by FIELD_5:12,FIELD_4:7;
   D4: Roots(E,p) c= the carrier of U by D3,D0,A,FIELD_8:27;
   Roots(E,p) = { Root1 p, Root2 p } by Z4; then
   Root1 p in Roots(E,p) by TARSKI:def 2; then
   reconsider w = Root1 p as Element of U by D4;
   sqrt(DC p) in the carrier of U
     proof
     consider aF being non zero Element of F,
              bF,cF being Element of F such that
     A1: p = <%cF,bF,aF%> by qua5;
     set aE = @(aF,E), bE = @(bF,E), cE = @(cF,E);
     C1: now assume C2: aE is zero;
         aF = 0.E by C2,FIELD_7:def 4 .= 0.F by K,C0SP1:def 3;
         hence contradiction;
         end;
         E is non 2-characteristic by K; then
     C3: 2 '*' aE <> 0.E by C1,ch2;
     I1: aE = aF & bE = bF & cE = cF by FIELD_7:def 4; then
         p = <%cE,bE,aE%> by A1,eval2; then
         Root1 p = (-bE + (RootDC p)) * (2 '*' aE)" by C1,Z2; then
        (Root1 p) * (2 '*' aE)
           = (-bE + (RootDC p)) * ((2 '*' aE)" * (2 '*' aE)) by GROUP_1:def 3
          .= (-bE + (RootDC p)) * 1.E by C3,VECTSP_1:def 10; then
     B: bE + (Root1 p) * (2 '*' aE)
           = (bE + -bE) + (RootDC p) by RLVECT_1:def 3
          .= 0.E + RootDC p by RLVECT_1:5;
     set aU = @(aF,U), bU = @(bF,U), cU = @(cF,U);
     I2: aU = aF & bU = bF & cU = cF by FIELD_7:def 4;
         2 '*' aU = 2 '*' aE  by D3,I2,Z3,FIELD_7:def 4; then
         w * (2 '*' aU) = (Root1 p) * (2 '*' aE) by D3,FIELD_6:16; then
     bU + w * (2 '*' aU) = bE + (Root1 p) * (2 '*' aE) by D3,I1,I2,FIELD_6:15;
     hence thesis by B;
     end; then
   D1: {sqrt(DC p)} c= the carrier of U by TARSKI:def 1;
   D2: U is Subfield of embField(canHomP X^2-(DC p)) by D0,EC_PF_1:5;
   F is Subfield of U by FIELD_4:7;
   then E is Subfield of U by D1,D2,FIELD_6:37;
   hence U == E by D0,FIELD_7:def 2;
   end;
hence thesis by A,FIELD_8:def 1;
end;
