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 m102:
for F being non 2-characteristic Field,
    E being FieldExtension of F
for a being Element of F st not(ex b being Element of F st a = b^2)
for b being Element of E st b^2 = a
holds FAdj(F,{b}) is SplittingField of X^2-a & deg(FAdj(F,{b}),F) = 2
proof
let F be non 2-characteristic Field, E be FieldExtension of F;
let a be Element of F;
assume AS1: not(ex b being Element of F st a = b^2);
let b be Element of E;
assume AS2: b^2 = a;
reconsider a1 = a as square Element of E by AS2;
F is Subring of E by FIELD_4:def 1; then
H1: 0.E = 0.F & 1.E = 1.F & -a1 = -a by FIELD_6:17,C0SP1:def 3;
H0: b*(-b) = -(b*b) by VECTSP_1:8 .= -a1 by AS2,O_RING_1:def 1;
H2: X^2-a = <%-a1,-0.E,1.E%> by H1
         .= <%b*(-b),-(b+(-b)),1.E%> by H0,RLVECT_1:5
         .= rpoly(1,b) *' rpoly(1,-b) by lemred3z;
rpoly(1,b) is Ppoly of E & rpoly(1,-b) is Ppoly of E by RING_5:51;
then reconsider p = X^2-a as Ppoly of E by H2,RING_5:52;
H3: 1.E * p = X^2-a; then
H11: X^2-a splits_in E by FIELD_4:def 5;
H5: Roots rpoly(1,b) = {b} & Roots rpoly(1,-b) = {-b} by RING_5:18;
H9: Roots(E,X^2-a) = Roots X^2-a1 by H1,FIELD_7:13
                  .= {b} \/ {-b} by H2,H1,H5,UPROOTS:23
                  .= {b,-b} by ENUMSET1:1; then
X^2-a splits_in FAdj(F,{b,-b}) by H3,FIELD_8:30,FIELD_4:def 5; then
H: X^2-a splits_in FAdj(F,{b}) by ext1;
now let U be FieldExtension of F;
  assume AS: X^2-a splits_in U & U is Subfield of FAdj(F,{b}); then
  H8: FAdj(F,{b}) is FieldExtension of U &
  E is FieldExtension of FAdj(F,{b}) by FIELD_4:7; then
  H6: U is Subfield of E by FIELD_4:7;
  H7: F is Subfield of U by FIELD_4:7;
  Roots(E,X^2-a) = Roots(U,X^2-a) by H11,AS,H8,FIELD_8:29; then
  H10: b in Roots(U,X^2-a) by H9,TARSKI:def 2;
  now let o be object;
    assume o in {b};
    then o = b by TARSKI:def 1;
    hence o in the carrier of U by H10;
    end;
  then {b} c= the carrier of U;
  then FAdj(F,{b}) is Subfield of U by H6,H7,FIELD_6:37;
  hence U == FAdj(F,{b}) by AS,FIELD_7:def 2;
  end;
hence FAdj(F,{b}) is SplittingField of X^2-a by H,FIELD_8:def 1;
eval(X^2-a1,b) = 0.E by AS2,m105; then
H12: Ext_eval(X^2-a,b) = 0.E by H1,FIELD_4:26; then
reconsider b1 = b as F-algebraic Element of E by FIELD_6:43;
X^2-a is irreducible by naH2,AS1,O_RING_1:def 2; then
H11: MinPoly(b1,F) = NormPolynomial X^2-a by H12,FIELD_8:15
                  .= X^2-a by RING_4:24;
deg X^2-a = 2 by defquadr;
hence thesis by H11,FIELD_6:67;
end;
