
theorem
for F being infinite Field
for E being F-finite FieldExtension of F
holds E is F-simple iff IntermediateFields(E,F) is finite
proof
let F be infinite Field, E be F-finite FieldExtension of F;
consider T being finite F-algebraic Subset of E such that
A: E == FAdj(F,T) by FIELD_7:37;
B: now assume AS: IntermediateFields(E,F) is finite;
   H: now let E be F-finite FieldExtension of F;
      assume AS: IntermediateFields(E,F) is finite;
      let T be finite F-algebraic Subset of E;
      assume H0: card T = 2 & E == FAdj(F,T); then
      consider a,b being object such that
      H1: a <> b & T = {a,b} by CARD_2:60;
      a in {a,b} & b in {a,b} by TARSKI:def 2; then
      reconsider a,b as Element of E by H1;
      ex x,y being Element of F st x <> y &
        FAdj(F,{a+@(x,E)*b}) = FAdj(F,{a+@(y,E)*b}) &
        FAdj(F,{a+@(x,E)*b}) is Subfield of E
        proof
        set I = IntermediateFields(E,F);
        defpred Q[object,object] means
          ex z being Element of F st $1 = z & $2 = FAdj(F,{a+@(z,E)*b});
        C1: for z being Element of F ex y being Element of I st Q[z,y]
            proof
            let z be Element of F;
            F is Subfield of FAdj(F,{a+@(z,E)*b}) by FIELD_6:36; then
            reconsider U = FAdj(F,{a+@(z,E)*b}) as Element of I by defY;
            thus ex y being Element of I st Q[z,y]
              proof
              take U;
              thus thesis;
              end;
            end;
        consider f being Function of F,I such that
        C2: for z being Element of F holds Q[z,f.z] from FUNCT_2:sch 3(C1);
        C3: dom f is infinite by FUNCT_2:def 1;
            rng f c= I; then
        not f is one-to-one by AS,C3,CARD_1:59; then
        consider x,y being object such that
        C4: x in the carrier of F & y in the carrier of F & f.x = f.y & x <> y
            by FUNCT_2:19;
        reconsider x,y as Element of F by C4;
        take x,y;
        thus x <> y by C4;
        Q[x,f.x] & Q[y,f.y] by C2;
        hence FAdj(F,{a+@(x,E)*b}) = FAdj(F,{a+@(y,E)*b}) by C4;
        thus thesis;
        end;
      then consider x,y being Element of F such that
      H2: x <> y & FAdj(F,{a+@(x,E)*b}) = FAdj(F,{a+@(y,E)*b}) &
          FAdj(F,{a+@(x,E)*b}) is Subfield of E;
      H3: b in FAdj(F,{a+@(x,E)*b})
          proof
          F is Subfield of FAdj(F,{a+@(x,E)*b}) by FIELD_6:36; then
          the carrier of F c= the carrier of FAdj(F,{a+@(x,E)*b})
            by EC_PF_1:def 1; then
          reconsider x1 = x, y1 = y as Element of FAdj(F,{a+@(x,E)*b});
          {a+@(x,E)*b} is Subset of FAdj(F,{a+@(x,E)*b}) &
             a+@(x,E)*b in {a+@(x,E)*b} by FIELD_6:35,TARSKI:def 1; then
          reconsider u = a+@(x,E)*b as Element of FAdj(F,{a+@(x,E)*b});
          {a+@(y,E)*b} is Subset of FAdj(F,{a+@(y,E)*b}) &
             a+@(y,E)*b in {a+@(y,E)*b} by FIELD_6:35,TARSKI:def 1; then
          reconsider v = a+@(y,E)*b as Element of FAdj(F,{a+@(x,E)*b}) by H2;
          H6: @(x,E) - @(y,E) <> 0.E
              proof
              H7: x - y <> 0.F by H2,RLVECT_1:21;
              H8: F is Subring of E by FIELD_4:def 1;
              H9: x = @(x,E) & y = @(y,E) by FIELD_7:def 4;
              -y = -@(y,E) by H8,FIELD_6:17,FIELD_7:def 4; then
              @(x,E) - @(y,E) = x - y by H8,H9,FIELD_6:15;
              hence thesis by H7,H8,C0SP1:def 3;
              end;
          a + @(x,E) * b - (a + @(y,E) * b)
              = a + @(x,E) * b - a - @(y,E) * b by RLVECT_1:27
             .= @(x,E) * b + a + -a + -(@(y,E) * b)
             .= @(x,E) * b + a + -(@(y,E) * b) + -a by RLVECT_1:def 3
             .= @(x,E) * b + -(@(y,E) * b) + a + -a by RLVECT_1:def 3
             .= @(x,E) * b - @(y,E) * b + (a + -a) by RLVECT_1:def 3
             .= (@(x,E) * b + (-@(y,E)) * b) + (a + -a) by VECTSP_1:9
             .= (@(x,E) + -@(y,E)) * b + (a + -a) by VECTSP_1:def 3
             .= (@(x,E) + -@(y,E)) * b + 0.E by RLVECT_1:5; then
          H7: (a + @(x,E) * b - (a + @(y,E) * b)) * (@(x,E) - @(y,E))"
              = b * ((@(x,E) + -@(y,E)) * (@(x,E) - @(y,E))") by GROUP_1:def 3
             .= b * 1.E by H6,VECTSP_1:def 10;
          H8: FAdj(F,{a+@(x,E)*b}) is Subring of E by FIELD_5:12;
          H9: x1 = @(x,E) by FIELD_7:def 4;
          -y1 = -@(y,E) by H8,FIELD_6:17,FIELD_7:def 4; then
          H11: x1 - y1 = @(x,E) - @(y,E) by H8,H9,FIELD_6:15;
          @(x,E) - @(y,E) is non zero by H6; then
          H12: (x1 - y1)" = (@(x,E) - @(y,E))" by H11,FIELD_6:18;
          -v = -(a + @(y,E) * b) by H8,FIELD_6:17; then
          u - v = a + @(x,E) * b - (a + @(y,E) * b) by H8,FIELD_6:15; then
          (u - v) * (x1 - y1)"
             = (a + @(x,E) * b - (a + @(y,E) * b)) * (@(x,E) - @(y,E))"
               by H12,H8,FIELD_6:16;
          hence thesis by H7;
          end;
      H4: a in FAdj(F,{a+@(x,E)*b})
          proof
          F is Subfield of FAdj(F,{a+@(x,E)*b}) by FIELD_6:36; then
          the carrier of F c= the carrier of FAdj(F,{a+@(x,E)*b})
            by EC_PF_1:def 1; then
          reconsider y1 = y as Element of FAdj(F,{a+@(x,E)*b});
          {a+@(y,E)*b} is Subset of FAdj(F,{a+@(y,E)*b}) &
             a+@(y,E)*b in {a+@(y,E)*b} by FIELD_6:35,TARSKI:def 1; then
          reconsider v = a+@(y,E)*b as Element of FAdj(F,{a+@(x,E)*b}) by H2;
          reconsider b1 = b as Element of FAdj(F,{a+@(x,E)*b}) by H3;
          H5: a + @(y,E) * b - (@(y,E) * b)
                = a + (@(y,E) * b + -(@(y,E) * b)) by RLVECT_1:def 3
               .= a + 0.E by RLVECT_1:5;
          H6: FAdj(F,{a+@(x,E)*b}) is Subring of E by FIELD_5:12;
          y1 = @(y,E) by FIELD_7:def 4; then
          @(y,E) * b = y1 * b1 by H6,FIELD_6:16; then
          -(@(y,E) * b) = -(y1 * b1) by H6,FIELD_6:17; then
          v + -(y1 * b1) = a + @(y,E) * b - (@(y,E) * b) by H6,FIELD_6:15;
          hence thesis by H5;
          end;
      now let o be object;
          assume o in {a,b}; then
          per cases by TARSKI:def 2;
          suppose o = a;
            hence o in the carrier of FAdj(F,{a+@(x,E)*b}) by H4;
            end;
          suppose o = b;
            hence o in the carrier of FAdj(F,{a+@(x,E)*b}) by H3;
            end;
          end; then
      H5: {a,b} c= the carrier of FAdj(F,{a+@(x,E)*b});
      H6: F is Subfield of FAdj(F,{a+@(x,E)*b}) by FIELD_6:36;
      E is FieldExtension of FAdj(F,{a+@(x,E)*b}) by FIELD_4:7; then
      FAdj(F,{a,b}) is FieldExtension of FAdj(F,{a+@(x,E)*b})
        by H0,H1,FIELD_13:11; then
      FAdj(F,{a+@(x,E)*b}) == FAdj(F,{a,b}) by H6,H5,FIELD_6:37,FIELD_4:7;
      hence ex p being Element of E st FAdj(F,{p}) = FAdj(F,T) by H1;
      end;
   defpred P[Nat] means
     for E being F-finite FieldExtension of F
     st IntermediateFields(E,F) is finite
     for T being finite F-algebraic Subset of E
     st card T = $1 & E == FAdj(F,T)
     holds ex p being Element of E st FAdj(F,{p}) = FAdj(F,T);
   I: for k being Nat st for n being Nat st n < k holds P[n] holds P[k]
      proof
      let k be Nat;
      assume B0: for n being Nat st n < k holds P[n];
      k <= 2 implies k = 0 or ... or k = 2; then
      per cases;
      suppose B1: k = 0;
        now let E be F-finite FieldExtension of F,
                T be finite F-algebraic Subset of E;
          assume card T = k & E == FAdj(F,T); then
          T c= the carrier of F by B1; then
          B2: FAdj(F,T) == F by FIELD_7:3;
          reconsider F1 = F as FieldExtension of F by FIELD_4:6;
          set a = the F-algebraic Element of F1;
          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; then
          reconsider a1 = a as Element of E;
          B3: FAdj(F,{a1}) = FAdj(F,{a}) by FIELD_10:9;
          FAdj(F,{a}) == F by FIELD_7:3;
          hence ex p being Element of E st FAdj(F,{p}) = FAdj(F,T) by B2,B3;
          end;
        hence P[k];
        end;
      suppose B1: k = 1;
        now let E be F-finite FieldExtension of F,
                T be finite F-algebraic Subset of E;
          assume card T = k & E == FAdj(F,T); then
          consider a being object such that B2: T = {a} by B1,CARD_2:42;
          a in T by B2,TARSKI:def 1; then
          reconsider a1 = a as Element of E;
          FAdj(F,{a1}) = FAdj(F,T) by B2;
          hence ex p being Element of E st FAdj(F,{p}) = FAdj(F,T);
          end;
        hence P[k];
        end;
      suppose k = 2;
        hence P[k] by H;
        end;
      suppose B1: k > 2;
        now let E be F-finite FieldExtension of F;
          assume AS: IntermediateFields(E,F) is finite;
          let T be finite F-algebraic Subset of E;
          assume B2: card T = k & E == FAdj(F,T);
          set a = the Element of T;
          T <> {} & T c= the carrier of E by B1,B2; then
          reconsider a as Element of E;
          reconsider T1 = T \ {a} as finite F-algebraic Subset of E;
          reconsider k1 = k - 1 as Nat by B1;
          now let o be object;
            assume o in {a};
            then o = a & T <> {} by B1,B2,TARSKI:def 1;
            hence o in T;
            end; then
          {a} c= T; then
          B6: T = T \/ {a} by XBOOLE_1:12 .= T1 \/ {a} by XBOOLE_1:39;
          a in {a} by TARSKI:def 1; then
          not a in T1 by XBOOLE_0:def 5; then
          B9: card T = card T1 + 1 by B6,CARD_2:41;
          set E1 = FAdj(F,T1);
          reconsider T2 = T1 as finite F-algebraic Subset of E1 by FIELD_6:35;
          B11: E is E1-extending by FIELD_4:7; then
          B12: IntermediateFields(E1,F) c= IntermediateFields(E,F) by simp2;
          B10: FAdj(F,T2) = FAdj(F,T1) by B11,FIELD_13:19; then
          consider q1 being Element of E1 such that
          B8: FAdj(F,{q1}) = FAdj(F,T2) by B12,AS,B0,B2,B9,NAT_1:19;
          the carrier of E1 c= the carrier of E by EC_PF_1:def 1; then
          reconsider q = q1 as Element of E;
          B4: FAdj(F,T) = FAdj(F,{q}\/{a}) by B6,B8,B10,B11,FIELD_13:19,simp1
                       .= FAdj(F,{q,a}) by ENUMSET1:1;
          per cases;
          suppose q = a;
            then {q,a} = {a} by ENUMSET1:29;
            hence ex p being Element of E st FAdj(F,{p}) = FAdj(F,T) by B4;
            end;
          suppose q <> a;
            then card {q,a} = 2 by CARD_2:57;
            hence ex p being Element of E st FAdj(F,{p}) = FAdj(F,T)
              by H,B4,B2,AS;
            end;
          end;
        hence P[k];
        end;
      end;
   J: for n being Nat holds P[n] from NAT_1:sch 4(I);
   consider n being Nat such that K: card T = n;
   consider p being Element of E such that
   L: FAdj(F,{p}) = FAdj(F,T) by AS,A,J,K;
   thus E is F-simple by A,L;
   end;
now assume E is F-simple; then
  consider a being Element of E such that C0: E == FAdj(F,{a});
  set I = IntermediateFields(E,F);
  defpred Q[object,object] means
     ex F1 being Field,
        K being FieldExtension of F1,
        E1 being F1-extending FieldExtension of K,
        a1 being K-algebraic Element of E1
     st F1 = F & E1 = E & a1 = a & $1 = K & $2 = MinPoly(a1,K);
  C1: for z being Element of I
      ex y being Element of Polynom-Ring E st Q[z,y]
      proof
      let z be Element of I;
      consider K being strict Field such that
      C2: K = z & F is Subfield of K & K is Subfield of E by defY;
      reconsider K as FieldExtension of F by C2,FIELD_4:7;
      reconsider E1 = E as F-extending FieldExtension of K by C2,FIELD_4:7;
      reconsider a1 = a as K-algebraic Element of E1;
      thus ex y being Element of Polynom-Ring E st Q[z,y]
        proof
        E is FieldExtension of K by C2,FIELD_4:7; then
        the carrier of Polynom-Ring K c= the carrier of Polynom-Ring E
          by FIELD_4:10; then
        reconsider y = MinPoly(a1,K) as Element of Polynom-Ring E;
        take y;
        thus Q[z,y] by C2;
        end;
      end;
  consider f being Function of I,Polynom-Ring E such that
  C2: for z being Element of I holds Q[z,f.z] from FUNCT_2:sch 3(C1);
  C3: for K being FieldExtension of F, E1 being FieldExtension of K,
          a1 being K-algebraic Element of E1
      st K is strict & E1 = E & a1 = a holds f.K = MinPoly(a1,K)
      proof
      let K be FieldExtension of F, E1 be FieldExtension of K,
          a1 be K-algebraic Element of E1;
      assume C4: K is strict & E1 = E & a1 = a;
      F is Subfield of K & K is Subfield of E1 by FIELD_4:7; then
      K is Element of I by C4,simp3;
      then Q[K,f.K] by C2; then
      consider F1 being Field,
        K1 being FieldExtension of F1,
        E2 being F1-extending FieldExtension of K1,
        a2 being K1-algebraic Element of E2 such that
      C5: F1 = F & E2 = E & a2 = a & K1 = K & f.K1 = MinPoly(a2,K1);
      thus thesis by C4,C5;
      end;
  C4: f is one-to-one
      proof
      now let x,y be object;
      assume C5: x in I & y in I & f.x = f.y; then
      consider Kx being strict Field such that
      C6: Kx = x & F is Subfield of Kx & Kx is Subfield of E by defY;
      consider Ky being strict Field such that
      C7: Ky = y & F is Subfield of Ky & Ky is Subfield of E by C5,defY;
      reconsider Kx as FieldExtension of F by C6,FIELD_4:7;
      reconsider Eh = E as Kx-extending FieldExtension of F by C6,FIELD_4:7;
      Eh is F-finite; then
      reconsider Kx as F-finite FieldExtension of F by FIELD_7:31;
      reconsider Ky as FieldExtension of F by C7,FIELD_4:7;
      reconsider Eh = E as Ky-extending FieldExtension of F by C7,FIELD_4:7;
      Eh is F-finite; then
      reconsider Ky as F-finite FieldExtension of F by FIELD_7:31;
      reconsider Ex = E as F-finite F-extending FieldExtension of Kx
          by C6,FIELD_4:7;
      reconsider Ey = E as F-finite F-extending FieldExtension of Ky
          by C7,FIELD_4:7;
      reconsider ax = a as Kx-algebraic Element of Ex;
      reconsider ay = a as Ky-algebraic Element of Ey;
      the carrier of Polynom-Ring Kx c= the carrier of Polynom-Ring Ex &
      the carrier of Polynom-Ring Ky c= the carrier of Polynom-Ring Ey
        by FIELD_4:10; then
      reconsider p1 = MinPoly(ax,Kx), p2 = MinPoly(ay,Ky)
                            as Element of the carrier of Polynom-Ring E;
      C8: Kx == FAdj(F,Coeff MinPoly(ax,Kx)) &
          Ky == FAdj(F,Coeff MinPoly(ay,Ky)) by C0,simpmainhelp;
      C9: p1 = f.Ky by C5,C6,C7,C3 .= p2 by C3;
      C10: Coeff p1 = Coeff MinPoly(ax,Kx) &
           Coeff p2 = Coeff MinPoly(ay,Ky) by co; then
      FAdj(F,Coeff MinPoly(ax,Kx))
          = FAdj(F,Coeff p2) by C9,FIELD_13:19
         .= FAdj(F,Coeff MinPoly(ay,Ky)) by C10,FIELD_13:19;
      hence x = y by C6,C7,C8;
      end;
      hence thesis by FUNCT_2:19;
      end;
  the carrier of Polynom-Ring F c= the carrier of Polynom-Ring E
      by FIELD_4:10; then
  reconsider q = MinPoly(a,F) as Element of the carrier of Polynom-Ring E;
     q <> 0_.(F); then
     q <> 0_.(E) by FIELD_4:12; then
     reconsider q as non zero Element of the carrier of Polynom-Ring E
       by UPROOTS:def 5;
  C5: for p being Element of the carrier of Polynom-Ring E
      st p in rng f holds p divides q
      proof
      let o be Element of the carrier of Polynom-Ring E;
      assume o in rng f; then
      consider x being object such that
      C7: x in I & o = f.x by FUNCT_2:11;
      reconsider x as Element of I by C7;
      Q[x,f.x] by C2; then
      consider F1 being Field,
        K1 being FieldExtension of F1,
        E2 being F1-extending FieldExtension of K1,
        a2 being K1-algebraic Element of E2 such that
      C8: F1 = F & E2 = E & a2 = a & K1 = x & f.K1 = MinPoly(a2,K1);
      the carrier of Polynom-Ring F c= the carrier of Polynom-Ring K1
         by C8,FIELD_4:10; then
      reconsider pF = MinPoly(a,F) as
                             Element of the carrier of Polynom-Ring K1;
      Ext_eval(pF,a) = Ext_eval(MinPoly(a,F),a) by C8,FIELD_8:6
                    .= 0.E by FIELD_6:52; then
      consider r being Polynomial of K1 such that
      B: MinPoly(a2,K1) *' r = pF by C8,FIELD_6:53,RING_4:1;
      C: r is Element of the carrier of Polynom-Ring K1 by POLYNOM3:def 10;
      the carrier of Polynom-Ring K1 c= the carrier of Polynom-Ring E
        by C8,FIELD_4:10; then
      reconsider q2 = MinPoly(a2,K1), r2 = r
                        as Element of the carrier of Polynom-Ring E by C;
      q2 *' r2 = MinPoly(a2,K1) *' r by C8,FIELD_4:17;
      hence o divides q by C8,C7,B,RING_4:1;
      end;
  now let o be object;
    assume C6: o in rng f; then
    consider x being object such that
    C7: x in I & o = f.x by FUNCT_2:11;
    reconsider x as Element of I by C7;
    Q[x,f.x] by C2; then
    consider F1 being Field,
        K1 being FieldExtension of F1,
        E2 being F1-extending FieldExtension of K1,
        a2 being K1-algebraic Element of E2 such that
    C8: F1 = F & E2 = E & a2 = a & K1 = x & f.K1 = MinPoly(a2,K1);
    the carrier of Polynom-Ring K1 c= the carrier of Polynom-Ring E
       by C8,FIELD_4:10; then
    reconsider p = MinPoly(a2,K1) as Element of the carrier of Polynom-Ring E;
    C9: K1 is Subfield of E by C8,FIELD_4:7;
    LC p = LC MinPoly(a2,K1) by C8,FIELD_8:5
        .= 1.K1 by RATFUNC1:def 7
        .= 1.E by C9,EC_PF_1:def 1; then
    reconsider p as monic Element of the carrier of Polynom-Ring E
       by RATFUNC1:def 7;
    p divides q by C8,C7,C6,C5;
    hence o in { p where p is monic
            Element of the carrier of Polynom-Ring E : p divides q } by C8,C7;
    end; then
  rng f c= MonicDivisors q; then
  dom f is finite by C4,CARD_1:59;
  hence IntermediateFields(E,F) is finite by FUNCT_2:def 1;
  end;
hence thesis by B;
end;
