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 Z4:
Roots(FAdj(F,{sqrt(DC p)}),p) = { Root1 p, Root2 p }
proof
set E = FAdj(F,{sqrt(DC p)}), r1 = Root1 p, r2 = Root2 p;
reconsider q = @(LC p,E) * (X-r1) *' (X-r2) as
              Element of the carrier of Polynom-Ring E by POLYNOM3:def 10;
K: F is Subring of E by FIELD_4:def 1;
Y: Roots(E,p) = Roots(q) by Z5,FIELD_7:13;
Z: now assume Y1: @(LC p,E) is zero;
     LC p = @(LC p,E) by FIELD_7:def 4 .= 0.F by Y1,K,C0SP1:def 3;
     hence contradiction;
     end;
Roots rpoly(1,r1) = {r1} & Roots rpoly(1,r2) = {r2} by RING_5:18; then
Roots(rpoly(1,r1) *' rpoly(1,r2))
       = {r1} \/ {r2} by UPROOTS:23 .= {r1,r2} by ENUMSET1:1;
hence thesis by Y,Z,RING_5:19;
end;
