
theorem
for F being Field
for E being F-finite FieldExtension of F
holds E is F-simple iff ex a being Element of E st deg(a,F) = deg(E,F)
proof
let F be Field, E be F-finite FieldExtension of F;
A: now assume E is F-simple;
   then consider a being Element of E such that A1: E == FAdj(F,{a});
   deg(E,F) = deg(a,F) by A1,FIELD_7:5;
   hence ex a being Element of E st deg(a,F) = deg(E,F);
   end;
now assume ex a being Element of E st deg(a,F) = deg(E,F);
  then consider a being Element of E such that A2: deg(a,F) = deg(E,F);
  set K = FAdj(F,{a});
  reconsider E1 = E as K-finite F-extending FieldExtension of K
                                               by FIELD_4:7,FIELD_7:31;
  deg(K,F) * (deg(K,F)")
      = (deg(E1,K) * deg(K,F)) * (deg(K,F)") by A2,FIELD_7:30
     .= deg(E1,K) * (deg(K,F) / deg(K,F))
     .= deg(E1,K) * 1 by XCMPLX_1:60;
  then deg(E1,K) = deg(K,F) / deg(K,F) .= 1 by XCMPLX_1:60;
  hence E is F-simple by FIELD_7:8;
  end;
hence thesis by A;
end;
