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 m105:
for F being Field
for a,b being Element of F st b^2 = a holds eval(X^2-a,b) = 0.F
proof
let F be Field; let a,b be Element of F;
assume b^2 = a; then
A: - a = -(b * b) by O_RING_1:def 1 .= b * (-b) by VECTSP_1:8;
thus  eval(X^2-a,b)
    = eval(<%-a,-0.F,1.F%>,b)
   .= eval(<%b*(-b),-(b+-b),1.F%>,b) by A,RLVECT_1:5
   .= eval(rpoly(1,b) *' rpoly(1,-b),b) by lemred3z
   .= eval(rpoly(1,b),b) * eval(rpoly(1,-b),b) by POLYNOM4:24
   .= (b - b) * eval(X--b,b) by HURWITZ:29
   .= 0.F * eval(X--b,b) by RLVECT_1:15
   .= 0.F;
end;
