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

theorem Fi3a:
for a being non square Element of F
holds FAdj(F,{sqrt a}) is SplittingField of X^2-a
proof
let a be non square Element of F;
set E = FAdj(F,{sqrt a});
reconsider p = X^2-a as Element of the carrier of Polynom-Ring F;
reconsider b = sqrt a as Element of E by FIELD_7:def 5;
reconsider q = rpoly(1,b) *' rpoly(1,-b) as
                        Element of the carrier of Polynom-Ring E
   by POLYNOM3:def 10;
E is Subring of embField(canHomP X^2-a) by FIELD_4:def 1; then
H: -b = -(sqrt a) by FIELD_6:17;
I: X-b = X-(sqrt a) by FIELD_4:21;
K: X+b = X+(sqrt a) by H,FIELD_4:21;
rpoly(1,b) *' rpoly(1,-b)
       = (X-(sqrt a)) *' (X+(sqrt a)) by I,K,FIELD_4:17
      .= X^2-a by Fi1a; then
D: X^2-a = 1.E * q;
   rpoly(1,b) is Ppoly of E & rpoly(1,-b) is Ppoly of E by RING_5:51;
   then q is Ppoly of E by RING_5:52; then
A: X^2-a splits_in E by D,FIELD_4:def 5;
now let U be FieldExtension of F;
   assume D0: X^2-a splits_in U & U is Subfield of E; then
   D3: E is U-extending by FIELD_4:7;
   D4: Roots(E,X^2-a) c= the carrier of U by D3,D0,A,FIELD_8:27;
   Roots(E,p) = { sqrt a, -(sqrt a) } by Fi2a; then
   sqrt a in Roots(E,p) by TARSKI:def 2; then
   sqrt a in the carrier of U by D4; then
   D1: {sqrt a} c= the carrier of U by TARSKI:def 1;
   D2: U is Subfield of embField(canHomP X^2-a) 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 E == U by D0,FIELD_7:def 2;
   end;
hence thesis by A,FIELD_8:def 1;
end;
