
theorem
for F being Field,
    E being FieldExtension of F
holds E is F-finite iff
      ex T being finite F-algebraic Subset of E st E == FAdj(F,T)
proof
let F be Field, E be FieldExtension of F;
defpred P[Nat] means
  for F being Field
  for E being FieldExtension of F st deg(E,F) = $1
  ex T being finite F-algebraic Subset of E st E == FAdj(F,T);
IA: now let F be Field, E be FieldExtension of F;
  assume deg(E,F) = 1;
  then A: F == E by str1a;
  {} c= the carrier of E; then
  reconsider T = {} as finite Subset of E;
  T is F-algebraic; then
  reconsider T as finite F-algebraic Subset of E;
  {} c= the carrier of F;
  then FAdj(F,T) == F by Th1;
  then E == FAdj(F,T) by A;
  hence ex T being finite F-algebraic Subset of E st E == FAdj(F,T);
  end;
IS: now let k be Nat;
    assume AS: k >= 1 & (for n being Nat st n >= 1 & n < k holds P[n]);
    now let F be Field, E be FieldExtension of F;
      assume Z: deg(E,F) = k; then
      VecSp(E,F) is finite-dimensional by FIELD_4:def 7; then
      reconsider K = E as F-finite FieldExtension of F by FIELD_4:def 8;
      per cases by AS,XXREAL_0:1;
      suppose k = 1; hence
        ex T being finite F-algebraic Subset of E st E == FAdj(F,T) by Z,IA;
        end;
      suppose B0: k > 1;
      now assume B1: not ex a being Element of K st not a in the carrier of F;
        for o be object holds o in the carrier of E iff o in the carrier of F
          proof
          let o be object;
          now assume B: o in the carrier of F;
             F is Subfield of E by FIELD_4:7; then
             the carrier of F c= the carrier of E by EC_PF_1:def 1;
             hence o in the carrier of E by B;
             end;
          hence thesis by B1;
          end;
        hence contradiction by Z,B0,quah1,TARSKI:2;
        end; then
      consider a being Element of K such that C1: not a in the carrier of F;
      set p = MinPoly(a,F);
      C2: {a} is F-algebraic;
      C3: deg(FAdj(F,{a}),F) <= k & deg(FAdj(F,{a}),F) > 1
          proof
          K is FAdj(F,{a})-extending by FIELD_4:7;
          hence deg(FAdj(F,{a}),F) <= k by Z,FIELD_5:15;
          C5: {a} is Subset of the carrier of FAdj(F,{a}) by FIELD_6:35;
          a in {a} by TARSKI:def 1; then
          C6: a in the carrier of FAdj(F,{a}) by C5;
          deg(FAdj(F,{a}),F) + 1 > 0 + 1 by XREAL_1:6;
          then deg(FAdj(F,{a}),F) >= 1 by NAT_1:13;
          hence deg(FAdj(F,{a}),F) > 1 by C6,C1,quah1,XXREAL_0:1;
          end;
      deg p = deg(FAdj(F,{a}),F) by FIELD_6:67; then
      per cases by C3,XXREAL_0:1;
      suppose deg p = k;
        then K == FAdj(F,{a}) by Z,ft1;
        hence ex T being finite F-algebraic Subset of E st E == FAdj(F,T)
          by C2;
        end;
      suppose deg p < k;
        set K = FAdj(F,{a});
        reconsider E1 = E as K-finite F-extending FieldExtension of K
           by alg0,FIELD_4:7;
        reconsider j = deg(E1,K) as Element of NAT by ORDINAL1:def 12;
        j >= 1 & j < k
            proof
            j + 1 > 0 + 1 by XREAL_1:6;
            hence j >= 1 by NAT_1:13;
            D7: k = deg(E1,K) * deg(K,F) by Z,degmult;
            D6: j <= k by D7,NAT_1:24;
            now assume j = k;
              then k / k = (k * deg(K,F)) / k by Z,degmult
                        .= (k / k) * deg(K,F);
              then k / k = 1 * deg(K,F) by B0,XCMPLX_1:60;
              hence contradiction by C3,XCMPLX_1:60;
              end;
            hence j < k by D6,XXREAL_0:1;
            end; then
        consider T1 being finite K-algebraic Subset of E1 such that
        D3: E1 == FAdj(K,T1) by AS;
        reconsider T = T1 as Subset of E;
        reconsider a1 = a as Element of E;
        for b being Element of E1 st b in T \/ {a} holds b is F-algebraic;
        then
        reconsider T2 = {a1} \/ T as F-algebraic Subset of E by defTalg;
        FAdj(K,T1) = FAdj(F,T2) by ug1;
        hence ex T being finite F-algebraic Subset of E st E == FAdj(F,T)
          by D3;
        end;
      end;
      end;
    hence P[k];
    end;
I: for k being Nat st k >= 1 holds P[k] from NAT_1:sch 9(IS);
A: now assume E is F-finite;
   then reconsider k = deg(E,F) as Element of NAT by ORDINAL1:def 12;
   k  >= 0 + 1 by NAT_1:13;
   hence ex T being finite F-algebraic Subset of E st E == FAdj(F,T) by I;
   end;
now assume ex T being finite F-algebraic Subset of E st E == FAdj(F,T);
   then consider T being finite F-algebraic Subset of E such that
   B0: E == FAdj(F,T);
   VecSp(FAdj(F,T),F) is finite-dimensional by FIELD_4:def 8;
   then VecSp(E,F) is finite-dimensional by B0,str12;
   hence E is F-finite by FIELD_4:def 8;
   end;
hence thesis by A;
end;
