
theorem
for F being Field holds
F is algebraic-closed iff
      for p being monic Element of the carrier of Polynom-Ring F
      holds p is irreducible iff deg p = 1
proof
let R be Field;
now assume AS: for p being monic Element of the carrier of Polynom-Ring R
      holds p is irreducible iff deg p = 1;
  now let p be Polynomial of R;
    assume A: len p > 1;
    then A1: p <> 0_.(R) by POLYNOM4:3;
    A4: len p - 1 > 1 - 1 by A,XREAL_1:9;
    then deg p is Element of NAT by INT_1:3;
    then A3: deg p >= 1 by NAT_1:14,A4;
    defpred P[Nat] means
       for p being Polynomial of R st deg p = $1 holds p is with_roots;
    II: for k being Nat st k >= 1 &
        (for n being Nat st n >= 1 & n < k holds P[n]) holds P[k]
      proof
      let k be Nat;
      assume IAS: k >= 1 & for n being Nat st n >= 1 & n < k holds P[n];
      per cases by IAS,XXREAL_0:1;
      suppose IA1: k = 1;
      thus P[k]
        proof
        now let p be Polynomial of R;
          assume deg p = 1;
          then consider x,z being Element of R such that
          I1: x <> 0.R & p = x * rpoly(1,z) by HURWITZ:28;
          eval(p,z) = x * eval(rpoly(1,z),z) by I1,POLYNOM5:30
                   .= x * (z - z) by HURWITZ:29
                   .= x *0.R by RLVECT_1:15;
          hence p is with_roots by POLYNOM5:def 8,POLYNOM5:def 7;
          end;
        hence thesis by IA1;
        end;
      end;
      suppose IA1: k > 1;
      thus P[k]
        proof
        now let p be Polynomial of R;
          reconsider pp = p as Element of Polynom-Ring R by POLYNOM3:def 10;
          set q = NormPolynomial(p);
          reconsider qq = q as Element of Polynom-Ring R by POLYNOM3:def 10;
          assume K: deg p = k;
          then len p <> 0;
          then I3: deg q > 1 by K,IA1,POLYNOM5:57;
          p <> 0_.(R) by K,HURWITZ:20;
          then p is non zero by UPROOTS:def 5;
          then I4: qq is reducible by AS,I3;
          I5: pp <> 0_.(R) by K,HURWITZ:20;
          not p is Unit of Polynom-Ring R by T8,K,IA1; then
          consider r being Element of the carrier of Polynom-Ring R such that
          I6: r divides p & 1 <= deg r & deg r < deg p by I4,I5,T88b,thirr2;
          r <> 0_.(R) by I6,HURWITZ:20; then
          deg r is Element of NAT by T8b;
          then consider x being Element of R such that
          I8: x is_a_root_of r by K,I6,IAS,POLYNOM5:def 8;
         reconsider rr=r,ppp=p as Element of Polynom-Ring R by POLYNOM3:def 10;
          rr divides ppp by I6; then
          consider u being Element of the carrier of Polynom-Ring R such that
          I10: ppp = rr * u;
          p = r *'u by I10,POLYNOM3:def 10; then
          eval(p,x) = eval(r,x) * eval(u,x) by POLYNOM4:24
                   .= 0.R * eval(u,x) by I8,POLYNOM5:def 7
                   .= 0.R;
          hence p is with_roots by POLYNOM5:def 8,POLYNOM5:def 7;
          end;
        hence thesis;
        end;
      end; end;
    I: for k being Nat st 1 <= k holds P[k] from NAT_1:sch 9(II);
    deg p is Element of NAT by A1,T8b; then
    consider n being Nat such that H: n >= 1 & deg p = n by A3;
    thus p is with_roots by H,I;
    end;
  hence R is algebraic-closed by POLYNOM5:def 9;
  end;
hence thesis by thirr1;
end;
