
theorem alg1:
for F being Field,
    E being FieldExtension of F
for a being Element of E holds a is F-algebraic iff a is_integral_over F
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_integral_over F;
   then consider p being Polynomial of F such that
   A1: LC p = 1.F & Ext_eval(p,a) = 0.E;
   (0_.(F)).(len(0_.(F))-'1) = 0.F; then
   p <> 0_.(F) by A1,RATFUNC1:def 6; then
   A3: p <> 0.(Polynom-Ring F) by POLYNOM3:def 10;
   reconsider b = p as Element of Polynom-Ring F by POLYNOM3:def 10;
   g.p = 0.E by A1,ALGNUM_1:def 11;
   then b in {x where x is Element of Polynom-Ring F : g.x = 0.E};
   then b in ker g by VECTSP10:def 9;
   hence a is F-algebraic by A3,TARSKI:def 1;
   end;
now assume A0: a is F-algebraic;
   now assume AS: not ex p being Element of the carrier of Polynom-Ring F
                  st p in ker g & p <> 0.(Polynom-Ring F);
      for x being object holds x in ker g iff x = 0.(Polynom-Ring F)
         proof
         let x be object;
         now assume A2: x = 0.(Polynom-Ring F);
           then g.x = 0.E by RING_2:6;
           then x in {v where v is Element of Polynom-Ring F : g.v=0.E} by A2;
           hence x in ker g by VECTSP10:def 9;
           end;
         hence thesis by AS;
         end;
      hence contradiction by A0,TARSKI:def 1;
      end;
   then consider p being Element of the carrier of Polynom-Ring F such that
   A1: p in ker g & p <> 0.(Polynom-Ring F);
   p <> 0_.(F) by A1,POLYNOM3:def 10; then
   reconsider p as non zero Element of the carrier of Polynom-Ring F
      by UPROOTS:def 5;
   set q = NormPolynomial p;
   A2: LC q = 1.F by RATFUNC1:def 7;
   ker g = {v where v is Element of Polynom-Ring F : g.v = 0.E}
       by VECTSP10:def 9;
   then consider v being Element of the carrier of Polynom-Ring F such that
   A3: v = p & g.v = 0.E by A1;
   Ext_eval(p,a) = 0.E by A3,ALGNUM_1:def 11; then
   Ext_eval(q,a) = 0.E by pr1;
   hence a is_integral_over F by A2;
   end;
hence thesis by A;
end;
