
theorem
for F being Field,
    E being FieldExtension of F
for T being non empty finite F-algebraic Subset of E
holds FAdj(F,T) is F-normal iff
      for a being Element of T holds MinPoly(a,F) splits_in FAdj(F,T)
proof
let F be Field, E be FieldExtension of F,
    T be non empty finite F-algebraic Subset of E;
set V = FAdj(F,T);
A: now assume B: V is F-normal;
   now let a be Element of T;
     T is Subset of V & a in T by FIELD_6:35; then
     reconsider a1 = a as Element of V;
     E is V-extending by FIELD_4:7; then
     Ext_eval(MinPoly(a,F),a1)
       = Ext_eval(MinPoly(a,F),a) by FIELD_6:11
      .= 0.E by FIELD_6:52
      .= 0.V by EC_PF_1:def 1; then
     a1 is_a_root_of MinPoly(a,F),V by FIELD_4:def 2;
     hence MinPoly(a,F) splits_in V by B,FIELD_4:def 3;
     end;
   hence for a being Element of T holds MinPoly(a,F) splits_in V;
   end;
now assume BB:
   for a being Element of T holds MinPoly(a,F) splits_in V;
set M = the set of all MinPoly(a,F) where a is Element of T;
set x = the Element of T;
E: MinPoly(x,F) in M;
now let o be object;
  assume o in the set of all MinPoly(a,F) where a is Element of T;
  then consider a being Element of T such that
  F: o = MinPoly(a,F);
  thus o in the carrier of Polynom-Ring F by F;
  end; then
reconsider M as non empty Subset of the carrier of Polynom-Ring F
   by E,TARSKI:def 3;
defpred Q[object,object] means
  ex a being Element of T st a = $1 & $2 = MinPoly(a,F);
E1: for a being Element of T ex y being Element of M st Q[a,y]
    proof
    let a be Element of T;
    MinPoly(a,F) in the set of all MinPoly(a,F) where a is Element of T;
    then reconsider y = MinPoly(a,F) as Element of M;
    take y;
    thus Q[a,y];
    end;
    consider f being Function of T,M such that
E2: for a being Element of T holds Q[a,f.a] from FUNCT_2:sch 3(E1);
E3: dom f = T by FUNCT_2:def 1;
    rng f = M
      proof
      E4: now let o be object;
          assume o in M; then
          consider a being Element of T such that
          E5: o = MinPoly(a,F);
          Q[a,f.a] by E2;
          hence o in rng f by E5,E3,FUNCT_1:def 3;
          end;
      now let o be object;
          assume o in rng f; then
          consider a be object such that
          E4: a in dom f & f.a = o by FUNCT_1:def 3;
          reconsider a as Element of T by E4;
          Q[a,o] by E2,E4;
          hence o in M;
          end;
      hence thesis by E4,TARSKI:2;
      end; then
reconsider M as non empty finite Subset of the carrier of Polynom-Ring F;
rng(canFS M) c= M & M c= the carrier of Polynom-Ring F by FINSEQ_1:def 4; then
len canFS M = card M & rng(canFS M) c= the carrier of Polynom-Ring F
   by FINSEQ_1:93; then
reconsider G = canFS M as
   non empty FinSequence of the carrier of Polynom-Ring F by FINSEQ_1:def 4;
reconsider p = Product G as Element of the carrier of Polynom-Ring F;
F: ex i being Element of dom G st G.i = MinPoly(x,F)
   proof
   rng(canFS M) = M by FUNCT_2:def 3; then
   MinPoly(x,F) in rng(canFS M); then
   consider i being object such that
   H: i in dom(canFS M) & (canFS M).i = MinPoly(x,F) by FUNCT_1:def 3;
   thus thesis by H;
   end;
Product G is non constant
  proof
  now let i be Element of dom G;
    H1: G.i in rng G by FUNCT_1:def 3;
    rng(canFS M) c= M by FINSEQ_1:def 4; then
    G.i in M by H1; then
    consider a being Element of T such that
    H2: G.i = MinPoly(a,F);
    thus G.i <> 0_.F by H2;
    end;
  then H: deg p >= deg MinPoly(x,F) by F,lemNor1deg;
  deg MinPoly(x,F) > 0 by RING_4:def 4;
  hence thesis by H,RING_4:def 4;
  end; then
reconsider p = Product G
   as non constant Element of the carrier of Polynom-Ring F;
D: p splits_in V
   proof
   deg p > 0 by RING_4:def 4; then
   reconsider q = p as non constant Polynomial of F by RATFUNC1:def 2;
   now let i be Element of dom G, p be Polynomial of F;
     assume D1: p = G.i;
     rng G c= M & G.i in rng G by FUNCT_1:3,FINSEQ_1:def 4; then
     G.i in M; then
     consider a being Element of T such that
     D2: G.i = MinPoly(a,F);
     thus p is constant or p splits_in V by D1,D2,BB;
     end;
   then q splits_in V by lemNor1a;
   hence thesis;
   end;
now let K be FieldExtension of F;
  assume E1: p splits_in K & K is Subfield of V; then
  E2: F is Subfield of K & V is K-extending by FIELD_4:7; then
  E3: Roots(K,p) = Roots(V,p) by D,E1,FIELD_8:29;
  E4: Roots(V,p) = {a where a is Element of V : a is_a_root_of p,V}
      by FIELD_4:def 4;
  E8: E is V-extending & K is Subfield of E by E1,EC_PF_1:5,FIELD_4:7;
  T is Subset of K
    proof
    now let o be object;
      assume Y: o in T; then
      reconsider a = o as Element of T;
      T is Subset of V by FIELD_6:35; then
      reconsider a1 = a as Element of V by Y;
      rng G = M by FUNCT_2:def 3; then
      MinPoly(a,F) in rng G; then
      consider i being object such that
      E5: i in dom G & G.i = MinPoly(a,F) by FUNCT_1:def 3;
      Ext_eval(MinPoly(a,F),a) = 0.E by FIELD_6:52; then
      0.E = Ext_eval(p,a) by E5,lemNor1b
         .= Ext_eval(p,a1) by E8,FIELD_7:14; then
      Ext_eval(p,a1) = 0.V by EC_PF_1:def 1; then
      a1 is_a_root_of p,V by FIELD_4:def 2; then
      a1 in Roots(V,p) by E4;
      hence o in the carrier of K by E3;
      end;
    hence thesis by TARSKI:def 3;
    end; then
  V is Subfield of K by E2,E8,FIELD_6:37;
  hence K == V by E1,FIELD_7:def 2;
  end;
   then V is SplittingField of p by D,FIELD_8:def 1;
   hence V is F-normal;
   end;
hence thesis by A;
end;
