reserve
F for non 2-characteristic non quadratic_complete polynomial_disjoint Field;

theorem
for F being non 2-characteristic Field,
    p being quadratic Element of the carrier of Polynom-Ring F
holds F is SplittingField of p iff DC p is square
proof
let F be non 2-characteristic Field;
let p be quadratic Element of the carrier of Polynom-Ring F;
consider a being non zero Element of F, b,c being Element of F such that
A: p = <%c,b,a%> by qua5;
B: now assume DC p is square; then
   b^2 - 4 '*' a * c is square by A,defDC; then
   consider w being Element of F such that
C: w^2 = b^2 - 4 '*' a * c by O_RING_1:def 2;
   set r1 = (-b + w) * (2 '*' a)", r2 = (-b - w) * (2 '*' a)";
D: <%c,b,a%> = a * (X-r1) *' (X-r2) by C,lemred;
   rpoly(1,r1) is Ppoly of F & rpoly(1,r2) is Ppoly of F by RING_5:51; then
E: rpoly(1,r1) *' rpoly(1,r2) is Ppoly of F by RING_5:52;
F: F is FieldExtension of F by FIELD_4:6;
   now let E be FieldExtension of F;
     assume F1: p splits_in E & E is Subfield of F;
     F is Subfield of E by FIELD_4:7;
     hence E == F by F1,FIELD_7:def 2;
     end;
   hence F is SplittingField of p by E,F,A,D,FIELD_4:def 5,FIELD_8:def 1;
   end;
now assume F is SplittingField of p; then
   p splits_in F by FIELD_8:def 1; then
   consider x being non zero Element of F, q being Ppoly of F such that
G: p = x * q by FIELD_4:def 5;
   Roots p <> {} by G; then
   b^2 - 4 '*' a * c is square by A,lemeval2;
   hence DC p is square by A,defDC;
   end;
hence thesis by B;
end;
