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

theorem Fi2a:
for a being non square Element of F
holds Roots(FAdj(F,{sqrt a}),X^2-a) = { sqrt a, -(sqrt 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;
H: E is Subring of embField(canHomP X^2-a) by FIELD_4:def 1; then
H1: -b = -(sqrt a) by FIELD_6:17;
Roots rpoly(1,b) = {b} & Roots rpoly(1,-b) = {-b} by RING_5:18; then
A: Roots(rpoly(1,b) *' rpoly(1,-b))
     = {b} \/ {-b} by UPROOTS:23
    .= {b,-b} by ENUMSET1:1;
I: X-b = X-(sqrt a) by FIELD_4:21;
K: X+b = X+(sqrt a) by H1,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 Roots(E,p) = Roots(q) by FIELD_7:13;
hence thesis by A,H,FIELD_6:17;
end;
