
theorem
for F being Field,
    E being FieldExtension of F
for a being Element of E
holds a is F-transcendental iff RAdj(F,{a}),Polynom-Ring F are_isomorphic
proof
let F be Field, E being FieldExtension of F; let a be Element of E;
reconsider E1 = E as (Polynom-Ring F)-homomorphic Field;
reconsider g = hom_Ext_eval(a,F) as Homomorphism of (Polynom-Ring F),E1;
H: (Polynom-Ring F)/(ker g), Image g are_isomorphic by RING_2:15;
A: now assume a is F-transcendental;
   then (Polynom-Ring F)/(ker g), (Polynom-Ring F) are_isomorphic by RING_2:17;
   then (Polynom-Ring F), (Image g) are_isomorphic by H,RING_3:44;
   hence (Polynom-Ring F), RAdj(F,{a}) are_isomorphic by lemphi4;
   end;
now assume RAdj(F,{a}),Polynom-Ring F are_isomorphic;
   then Polynom-Ring F is (RAdj(F,{a}))-isomorphic by RING_3:def 4;
   then RAdj(F,{a}) <> FAdj(F,{a});

   hence a is F-transcendental by ch1;
   end;
hence thesis by A;
end;
