reserve m,n for Nat;
reserve r for Real;
reserve c for Element of F_Complex;

theorem Th25:
  c is algebraic iff
  ex f being monic RAT-valued Polynomial of F_Complex st c is_a_root_of f
  proof
    hereby
      assume c is algebraic;
      then c is_integral_over Fq;
      then consider f being Polynomial of Fq such that
A1:   LC f = 1.Fq and
A2:   Ext_eval(f,c) = 0.FC;
      reconsider f1 = f as RAT-valued Polynomial of FC by ALGNUM_1:3,Th8;
      f1 is monic by A1,Th9,Lm2,ALGNUM_1:3;
      then reconsider f1 as monic RAT-valued Polynomial of FC;
      take f1;
      thus c is_a_root_of f1
      proof
        thus eval(f1,c) = In(eval(f1,c),FC)
        .= Ext_eval(f1,In(c,FC)) by Lm3,ALGNUM_1:12
        .= 0.FC by A2,Th15,ALGNUM_1:3;
      end;
    end;
    given f1 being monic RAT-valued Polynomial of FC such that
A3: c is_a_root_of f1;
    reconsider f = f1 as Polynomial of Fq by Th21;
    take f;
    LC f1 = 1.FC by RATFUNC1:def 7;
    hence LC f = 1.Fq by Th9,Lm2,ALGNUM_1:3;
    thus Ext_eval(f,c) = Ext_eval(f1,In(c,FC)) by Th15,ALGNUM_1:3
    .= In(eval(f1,c),FC) by Lm3,ALGNUM_1:12
    .= 0.FC by A3;
  end;
