
theorem mpol4:
for F being Field,
    E being FieldExtension of F
for a being F-algebraic Element of E,
    p being Element of the carrier of Polynom-Ring F
holds Ext_eval(p,a) = 0.E iff MinPoly(a,F) divides p
proof
let F be Field, E be FieldExtension of F;
let a be F-algebraic Element of E,
    p be Element of the carrier of Polynom-Ring F;
set ma = MinPoly(a,F), g = hom_Ext_eval(a,F);
X: F is Subring of E by FIELD_4:def 1;
reconsider p1 = p, ma1 = ma as Element of Polynom-Ring F;
A: now assume Ext_eval(p,a) = 0.E; then
   g.p = 0.E by ALGNUM_1:def 11; then
   p in {v where v is Element of Polynom-Ring F : g.v = 0.E}; then
   p in ker g by VECTSP10:def 9; then
   p in {ma}-Ideal by mpol1;
   hence ma divides p by RING_4:def 3,RING_2:18;
   end;
now assume ma divides p;
   then consider u being Polynomial of F such that H: ma *' u = p by RING_4:1;
   0.E = Ext_eval(ma,a) by mpol2;
   hence 0.E = Ext_eval(ma,a) * Ext_eval(u,a)
            .= Ext_eval(p,a) by X,H,ALGNUM_1:20;
   end;
hence thesis by A;
end;
