
theorem quah1:
for F being Field
for E being FieldExtension of F
holds deg(E,F) = 1 iff the carrier of E = the carrier of F
proof
let F be Field; let E be FieldExtension of F;
H0: F is Subring of E & F is Subfield of E by FIELD_4:def 1,FIELD_4:7; then
A: the carrier of F c= the carrier of E & 1.E = 1.F by C0SP1:def 3;
set V = VecSp(E,F);
reconsider e = 1.E as Element of V by FIELD_4:def 6;
Z: now assume AS: deg(E,F) = 1; then
   H1: V is finite-dimensional by FIELD_4:def 7;
   dim V = 1 by AS,FIELD_4:def 7; then
   consider v being Element of V such that
   B: v <> 0.V & (Omega).V = Lin {v} by H1,VECTSP_9:30;
   C: the carrier of Lin {v} = the carrier of V by B,VECTSP_4:def 4;
   reconsider a = v as Element of E by FIELD_4:def 6;
   e in Lin {v} by C; then
   consider l being Linear_Combination of {v} such that
   D2: e = Sum l by VECTSP_7:7;
   reconsider xF = l.v as Element of F;
   reconsider xE = xF as Element of E by A;
   D3: [xE,a] in [:the carrier of F,the carrier of E:] by ZFMISC_1:def 2;
   D4: 1.E = l.v * v by D2,VECTSP_6:17
     .= (the multF of E)|[:the carrier of F,the carrier of E:].(xF,a)
        by FIELD_4:def 6
     .= xE * a by D3,FUNCT_1:49; then
   D5: xE <> 0.E;
   xE is non zero by D4; then
   xF" * 1.F = xE" * 1.E by H0,FIELD_6:18
           .= (xE" * xE) * a by D4,GROUP_1:def 3
           .= 1.E * a by D5,VECTSP_1:def 10; then
   reconsider vF =  v as Element of F;
   now let o be object;
     assume o in the carrier of E; then
     o in Lin {v} by C,FIELD_4:def 6; then
     consider l being Linear_Combination of {v} such that
     E: o = Sum l by VECTSP_7:7;
     reconsider x = l.v as Element of F;
     F: [x,a] in [:the carrier of F,the carrier of E:] by ZFMISC_1:def 2;
     G: [x,vF] in [:the carrier of F,the carrier of F:] by ZFMISC_1:def 2;
     o = l.v * v by E,VECTSP_6:17
      .= (the multF of E)|[:the carrier of F,the carrier of E:].(x,vF)
         by FIELD_4:def 6
      .= (the multF of E).(x,vF) by F,FUNCT_1:49
      .= ((the multF of E)||(the carrier of F)).(x,vF) by G,FUNCT_1:49
      .= x * vF by H0,C0SP1:def 3;
     hence o in the carrier of F;

     end;
   hence the carrier of E = the carrier of F by A,TARSKI:2;
   end;
now assume AS1: the carrier of E = the carrier of F;
reconsider A = {e} as Subset of V;
0.V = 0.E by FIELD_4:def 6;
then L: A is linearly-independent by VECTSP_7:3;
H1: the carrier of Lin(A) =
the set of all Sum(l) where l is Linear_Combination of A by VECTSP_7:def 2;

H2: now let o be object;
    assume o in the carrier of V;
    then reconsider v = o as Element of the carrier of V;
    reconsider a = v as Element of E by FIELD_4:def 6;
    defpred P[object,object] means ($1 = e & $2 = a) or ($1 <> e & $2 = 0.E);
    G0: for x being object st x in the carrier of V
        ex y being object st y in the carrier of E & P[x,y]
        proof
        let o be object;
        assume o in the carrier of V;
        per cases;
        suppose A: o = e; take a; thus thesis by A; end;
        suppose A: o <> e; take 0.E; thus thesis by A; end;
        end;
    consider f being Function of the carrier of V, the carrier of E such that
    G1: for x being object st x in the carrier of V holds P[x,f.x]
        from FUNCT_2:sch 1(G0);
    dom f = the carrier of V & rng f c= the carrier of E by FUNCT_2:def 1; then
    reconsider f as Element of Funcs(the carrier of V, the carrier of F)
       by AS1,FUNCT_2:def 2;
    ex T being finite Subset of V st
    for v being Element of V st not v in T holds f.v = 0.F
      proof
      reconsider T = {e} as finite Subset of V;
      take T;
      now let u be Element of V;
        assume not u in T;
        then u <> e by TARSKI:def 1;
        hence f.u = 0.E by G1 .= 0.F by H0,C0SP1:def 3;
        end;
      hence thesis;
      end;
    then reconsider l = f as Linear_Combination of V by VECTSP_6:def 1;
    now let o be object;
      assume o in Carrier l; then
      o in {v where v is Element of V: l.v <> 0.F} by VECTSP_6:def 2;
      then consider u being Element of V such that
      I: o = u & l.u <> 0.F;
      l.u <> 0.E by I,H0,C0SP1:def 3; then
      u = e by G1;
      hence o in A by I,TARSKI:def 1;
      end;
    then Carrier l c= A;
    then reconsider l as Linear_Combination of A by VECTSP_6:def 4;
    Sum l = l.e * e by VECTSP_6:17
         .= ((the multF of E)|[:the carrier of F,the carrier of E:]).(l.e,e)
            by FIELD_4:def 6
         .= a * 1.E by AS1,G1
         .= v;
    hence o in the set of all Sum(l) where l is Linear_Combination of A;
    end;

now let o be object;
  assume o in the set of all Sum(l) where l is Linear_Combination of A;
  then consider l being Linear_Combination of A such that G: o = Sum l;
  thus o in the carrier of V by G;
  end;
then the carrier of V = the set of all Sum(l) where
                              l is Linear_Combination of A by H2,TARSKI:2;
then M: {1.E} is Basis of VecSp(E,F) by H1,L,VECTSP_4:31,VECTSP_7:def 3;
  H4: card {1.E} = 1 by CARD_1:30;
  H5: V is finite-dimensional by M,MATRLIN:def 1; then
  dim V = 1 by M,H4,VECTSP_9:def 1;
  hence deg(E,F) = 1 by H5,FIELD_4:def 7;
  end;
hence thesis by Z;
end;
