
theorem ch1:
for F being Field,
    E being (Polynom-Ring F)-homomorphic FieldExtension of F,
    a being Element of E holds a is F-algebraic iff FAdj(F,{a}) = RAdj(F,{a})
proof
let F be Field, E be (Polynom-Ring F)-homomorphic FieldExtension of F,
    a be Element of E;
set f = hom_Ext_eval(a,F);
A: now assume a is F-algebraic;
   then reconsider b = a as F-algebraic Element of E;
   (Polynom-Ring F)/({MinPoly(b,F)}-Ideal) is Field;
   then A1: (Polynom-Ring F)/(ker f) is Field by mpol1;
   (Polynom-Ring F)/(ker f), Image f are_isomorphic by RING_2:15; then
   reconsider K = Image f as ((Polynom-Ring F)/(ker f))-isomorphic Ring
      by RING_3:def 4;
   K is Field by A1; then
   RAdj(F,{a}) is Field by lemphi4;
   hence FAdj(F,{a}) = RAdj(F,{a}) by RF2;
   end;
now assume FAdj(F,{a}) = RAdj(F,{a});
   then B1: Image f is Field by lemphi4;
   (Image f),(Polynom-Ring F)/(ker f) are_isomorphic by RING_2:15; then
   reconsider K = (Polynom-Ring F)/(ker f) as (Image f)-isomorphic Ring
      by RING_3:def 4;
   B2: K is Field by B1;
   {0.(Polynom-Ring F)}-Ideal = {0.(Polynom-Ring F)} by IDEAL_1:47;
   hence a is F-algebraic by B2,RING_1:21;
   end;
hence thesis by A;
end;
