
theorem alg00:
for F being Field,
    E being FieldExtension of F
for a being Element of E
holds a is F-algebraic iff
      ex p being non zero Polynomial of F st Ext_eval(p,a) = 0.E
proof
let F be Field, E being FieldExtension of F; let a be Element of E;
set g = hom_Ext_eval(a,F);
A: now assume a is F-algebraic;
   then a is_integral_over F by alg1;
   then consider p being Polynomial of F such that
   A1: LC p = 1.F & Ext_eval(p,a) = 0.E;
   LC(0_.(F)) = (0_.(F)).(len (0_.(F))-'1) by RATFUNC1:def 6
             .= 0.F;
   then p is non zero by A1,UPROOTS:def 5;
   hence ex p being non zero Polynomial of F st Ext_eval(p,a) = 0.E by A1;
   end;
now assume ex p being non zero Polynomial of F st Ext_eval(p,a) = 0.E;
   then consider p being non zero Polynomial of F such that
   A1: Ext_eval(p,a) = 0.E;
   A2: g.p = 0.E by A1,ALGNUM_1:def 11;
   reconsider b = p as Element of Polynom-Ring F by POLYNOM3:def 10;
   p <> 0_.(F);
   then b is non zero by POLYNOM3:def 10; then
   reconsider b as non zero Element of Polynom-Ring F;
   b in {x where x is Element of Polynom-Ring F : g.x = 0.E} by A2;
   then b in ker g by VECTSP10:def 9;
   hence a is F-algebraic by TARSKI:def 1;
   end;
hence thesis by A;
end;
