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
for E being FieldExtension of F holds
E is F-quadratic iff
ex a being non square Element of F st E,FAdj(F,{sqrt a}) are_isomorphic_over F
proof
let E be FieldExtension of F;
H1: F is Subring of E & F is Subfield of E by FIELD_4:def 1,FIELD_4:7; then
H2: the carrier of F c= the carrier of E by C0SP1:def 3;
A:now assume AS: E is F-quadratic; then
  reconsider K = E as F-finite FieldExtension of F;
  now assume not ex a being Element of K st not a in the carrier of F;
    then the carrier of K c= the carrier of F;
    hence contradiction by AS,H2,TARSKI:2,FIELD_7:7;
    end; then
  consider a being Element of K such that A: not a in the carrier of F;
  B: deg(FAdj(F,{a}),F) = 2
     proof
     K is FAdj(F,{a})-extending by FIELD_4:7; then
     C1: deg(FAdj(F,{a}),F) <= 2 by AS,FIELD_5:15;
     now assume deg(FAdj(F,{a}),F) < 1 + 1; then
       C2: deg(FAdj(F,{a}),F) <= 1 by NAT_1:13;
       deg(FAdj(F,{a}),F) + 1 > 0 + 1 by XREAL_1:6; then
       deg(FAdj(F,{a}),F) >= 1 by NAT_1:13; then
       C3: the carrier of FAdj(F,{a}) = the carrier of F
           by C2,XXREAL_0:1,FIELD_7:7;
       C4: {a} is Subset of FAdj(F,{a}) by FIELD_6:35;
       a in {a} by TARSKI:def 1;
       hence contradiction by A,C3,C4;
       end;
     hence thesis by C1,XXREAL_0:1;
     end; then
  deg MinPoly(a,F) = 2 by FIELD_6:67; then
  reconsider p = MinPoly(a,F) as
             quadratic Element of the carrier of Polynom-Ring F by defquadr;

  D: K, FAdj(F,{a}) are_isomorphic_over F
     proof
     D1: VecSp(K,F) is finite-dimensional by FIELD_4:def 8;
     D2: dim VecSp(K,F) = 2 by AS,FIELD_4:def 7
                   .= dim VecSp(FAdj(F,{a}),F) by B,FIELD_4:def 7;
     D3: K is FAdj(F,{a})-extending by FIELD_4:7;
     VecSp(FAdj(F,{a}),F) is Subspace of VecSp(K,F) by D3,FIELD_5:14; then
     (Omega).VecSp(K,F) = (Omega).VecSp(FAdj(F,{a}),F) by D1,D2,VECTSP_9:28;
     then
     D4: the carrier of VecSp(K,F)
            = the carrier of (Omega).VecSp(FAdj(F,{a}),F) by VECTSP_4:def 4
           .= the carrier of VecSp(FAdj(F,{a}),F) by VECTSP_4:def 4
           .= the carrier of FAdj(F,{a}) by FIELD_4:def 6; then
     D8: the carrier of K = the carrier of FAdj(F,{a}) by FIELD_4:def 6;
     D6: K is Subfield of FAdj(F,{a})
       proof
       the addF of FAdj(F,{a}) = (the addF of K)||the carrier of FAdj(F,{a}) &
       the multF of FAdj(F,{a}) = (the multF of K)||the carrier of FAdj(F,{a})
         & 1.FAdj(F,{a}) = 1.K & 0.FAdj(F,{a}) = 0.K by EC_PF_1:def 1;
       hence thesis by D8,EC_PF_1:def 1;
       end;
     reconsider f = id K as Function of K,FAdj(F,{a}) by D4,FIELD_4:def 6;
     D7: f is isomorphism by D6,unique20;
     now let a be Element of F;
        reconsider a1 = a as Element of E by H2;
        f.a1 = a1;
        hence f.a = a;
        end;
     then f is F-fixing;
     hence thesis by D7,FIELD_8:def 5;
     end;

  consider b,c being Element of F such that E: p = <%c,b,1.F%> by qua5a;
  reconsider b1 = b, c1 = c as Element of K by H2;
  the carrier of Polynom-Ring F c= the carrier of Polynom-Ring K by FIELD_4:10;
  then reconsider p1 = p as Element of the carrier of Polynom-Ring K;

     1.F = 1.K by H1,C0SP1:def 3; then
  F: p = <%c1,b1,1.K%> by E,eval2;
  0.K = Ext_eval(p,a) by FIELD_6:51 .= eval(p1,a) by FIELD_4:26; then
  a is_a_root_of p1; then
  G: a in Roots p1 by POLYNOM5:def 10;
  G1: b1^2 = b1 * b1 by O_RING_1:def 1
          .= b * b by H1,FIELD_6:16 .= b^2 by O_RING_1:def 1;
  4 '*' c1 = 4 '*' c by Z3; then
      - (4 '*' c1) = - (4 '*' c) by H1,FIELD_6:17; then
  U2: b1^2 - 4 '*' c1 = b^2 - 4 '*' c by G1,H1,FIELD_6:15;
  I: b1^2 - 4 '*' 1.K * c1
          = b1^2 - 4 '*' (1.K * c1) by REALALG2:5
         .= b1^2 - 4 '*' c1; then
  consider u being Element of K such that
  U1: u^2 = b1^2 - 4 '*' c1 by G,F,lemeval2,O_RING_1:def 2;

  L0: K is non 2-characteristic by H1; then
  L1: Roots <%c1,b1,1.K%> = {(-b1+u) * (2'*'1.K)", (-b1-u) * (2'*'1.K)" }
      by I,U1,TC0;
  G5: 2 '*' 1.K is non zero by L0,ch2;
  0.K = Ext_eval(p,a) by FIELD_6:51 .= eval(p1,a) by FIELD_4:26; then
  a is_a_root_of p1; then
  L2: a in Roots p1 by POLYNOM5:def 10;
  L3: u = b1 + a * (2 '*' 1.K) or u = -(b1 + a * (2 '*' 1.K))
      proof
      per cases by L2,F,L1,TARSKI:def 2;
      suppose a = (-b1 + u) * (2 '*' 1.K)";
        then a * (2 '*' 1.K)
           = (-b1 + u) * ((2 '*' 1.K)" * (2 '*' 1.K)) by GROUP_1:def 3
          .= (-b1 + u) * 1.K by G5,VECTSP_1:def 10;
        then b1 + a * (2 '*' 1.K)
           = (b1 + -b1) + u by RLVECT_1:def 3
          .= 0.K + u by RLVECT_1:5;
        hence thesis;
        end;
      suppose a = (-b1 - u) * (2 '*' 1.K)";
        then a * (2 '*' 1.K)
           = (-b1 - u) * ((2 '*' 1.K)" * (2 '*' 1.K)) by GROUP_1:def 3
          .= (-b1 - u) * 1.K by G5,VECTSP_1:def 10;
        then b1 + a * (2 '*' 1.K)
           = (b1 + -b1) - u by RLVECT_1:def 3
          .= 0.K - u by RLVECT_1:5;
        hence thesis;
        end;
      end;

  U3: FAdj(F,{a}) = FAdj(F,{u})
      proof
      G2: K is FAdj(F,{a})-extending &
          K is FAdj(F,{u})-extending by FIELD_4:7; then
      G3: F is Subfield of FAdj(F,{u}) & F is Subfield of FAdj(F,{a}) &
          FAdj(F,{u}) is Subring of K & FAdj(F,{a}) is Subring of K
          by FIELD_6:36,FIELD_4:def 1; then
      I1: the carrier of F c= the carrier of FAdj(F,{u}) &
          the carrier of F c= the carrier of FAdj(F,{a}) by EC_PF_1:def 1;
      I2: {u} is Subset of FAdj(F,{u}) & {a} is Subset of FAdj(F,{a})
          by FIELD_6:35;
      u in {u} by TARSKI:def 1; then
      reconsider b2 = b, c2 = c, u2 = u as Element of FAdj(F,{u}) by I1,I2;
      a = (-b2 + u2) * (2 '*' 1.FAdj(F,{u}))" or
          a = (-b2 - u2) * (2 '*' 1.FAdj(F,{u}))"
        proof
        1.K = 1.FAdj(F,{u}) by G3,C0SP1:def 3; then
        2 '*' 1.K = 2 '*' 1.FAdj(F,{u}) by G2,Z3; then
        G4: (2 '*' 1.K)" = (2 '*' 1.FAdj(F,{u}))" by G5,FIELD_6:18;
        G6: -b1 = -b2 & -u = -u2 by G3,FIELD_6:17;
        per cases by L2,F,L1,TARSKI:def 2;
        suppose K: a = (-b1 + u) * (2 '*' 1.K)";
          (-b1 + u) = (-b2 + u2) by G6,G3,FIELD_6:15;
          hence thesis by K,G3,G4,FIELD_6:16;
          end;
        suppose K: a = (-b1 - u) * (2 '*' 1.K)";
          (-b1 - u) = (-b2 - u2) by G6,G3,FIELD_6:15;
          hence thesis by K,G3,G4,FIELD_6:16;
          end;
        end; then
      a in the carrier of FAdj(F,{u}); then
      {a} c= the carrier of FAdj(F,{u}) by TARSKI:def 1; then
      G4: FAdj(F,{a}) is Subfield of FAdj(F,{u}) by G3,FIELD_6:37;
      a in {a} by TARSKI:def 1; then
      reconsider b2 = b, c2 = c, a2 = a as Element of FAdj(F,{a}) by I1,I2;
      u = b2 + a2 * (2 '*' 1.FAdj(F,{a})) or
          u = -(b2 + a2 * (2 '*' 1.FAdj(F,{a})))
        proof
        1.K = 1.FAdj(F,{a}) by G3,C0SP1:def 3; then
        2 '*' 1.K = 2 '*' 1.FAdj(F,{a}) by G2,Z3; then
        G6: a * (2 '*' 1.K) = a2 * (2 '*' 1.FAdj(F,{a})) by G3,FIELD_6:16; then
        G5: b1 + a * (2 '*' 1.K) = b2 + a2 * (2 '*' 1.FAdj(F,{a}))
            by G3,FIELD_6:15;
        per cases by L3;
        suppose u = b1 + a * (2 '*' 1.K);
          hence thesis by G6,G3,FIELD_6:15;
          end;
        suppose u = -(b1 + a * (2 '*' 1.K));
          hence thesis by G3,G5,FIELD_6:17;
          end;
        end; then
      u in the carrier of FAdj(F,{a}); then
      {u} c= the carrier of FAdj(F,{a}) by TARSKI:def 1; then
      FAdj(F,{u}) is Subfield of FAdj(F,{a}) by G3,FIELD_6:37;
      hence FAdj(F,{a}) = FAdj(F,{u}) by G4,EC_PF_1:4;
      end;

  U4: now assume ex w being Element of F st b^2 - 4 '*' c = w^2; then
      consider w being Element of F such that G7: b^2 - 4 '*' c = w^2;
          1.F = 1.K by H1,C0SP1:def 3; then
      F2: 2 '*' 1.F = 2 '*' 1.K by Z3;
          2 '*' 1.K is non zero by L0,ch2; then
      F3: (2 '*' 1.F)" = (2 '*' 1.K)" by H1,F2,FIELD_6:18;
      F4: -b1 = -b by H1,FIELD_6:17;
      per cases by G7,U1,U2,lemquadr;
      suppose S: u = w; then
        -b1 + u = -b + w by H1,F4,FIELD_6:15; then
        F5: (-b1 + u) * (2 '*' 1.K)" = (-b + w) * (2 '*' 1.F)"
            by H1,F3,FIELD_6:16;
        -u = -w by H1,S,FIELD_6:17; then
        (-b1 - u) = (-b - w) by H1,F4,FIELD_6:15; then
        F6: (-b1 - u) * (2 '*' 1.K)" = (-b - w) * (2 '*' 1.F)"
            by H1,F3,FIELD_6:16;
        per cases by L2,F,L1,TARSKI:def 2;
        suppose a = (-b1 + u) * (2 '*' 1.K)";
          hence contradiction by F5,A;
          end;
        suppose a = (-b1 - u) * (2 '*' 1.K)";
          hence contradiction by F6,A;
          end;
        end;
      suppose S: u = -w;then
        -b1 + u = -b + (-w) by H1,F4,FIELD_6:15; then
        F5: (-b1 + u) * (2 '*' 1.K)" = (-b + (-w)) * (2 '*' 1.F)"
            by H1,F3,FIELD_6:16;
        -u = --w by H1,S,FIELD_6:17; then
        (-b1 - u) = (-b + w) by H1,F4,FIELD_6:15; then
        F6: (-b1 - u) * (2 '*' 1.K)" = (-b + w) * (2 '*' 1.F)"
            by H1,F3,FIELD_6:16;
        per cases by L2,F,L1,TARSKI:def 2;
        suppose a = (-b1 + u) * (2 '*' 1.K)";
          hence contradiction by F5,A;
          end;
        suppose a = (-b1 - u) * (2 '*' 1.K)";
          hence contradiction by F6,A;
          end;
        end;
      end; then

  reconsider v = b^2 - 4'*'c as non square Element of F by O_RING_1:def 2;
  F5: FAdj(F,{sqrt v}) is SplittingField of (X^2-v) by Fi3a;
  F6: FAdj(F,{a}) is SplittingField of (X^2-v) by U1,U2,U3,U4,m102;
  FAdj(F,{a}),FAdj(F,{sqrt v}) are_isomorphic_over F by F5,F6,FIELD_8:58;
  hence ex a being non square Element of F
        st E,FAdj(F,{sqrt a}) are_isomorphic_over F by D,FIELD_8:44;
end;
now given a being non square Element of F such that
  A: E,FAdj(F,{sqrt a}) are_isomorphic_over F;
  FAdj(F,{sqrt a}),E are_isomorphic_over F by A,FIELD_8:43;
  hence deg(E,F) = deg(FAdj(F,{sqrt a}),F) by FIELD_8:45 .= 2 by dega;
  end;
hence thesis by A;
end;
