
theorem
for F be Field
for E be F-finite FieldExtension of F
for a being Element of E holds a is F-primitive iff deg(a,F) = deg(E,F)
proof
let F be Field, E be F-finite FieldExtension of F, a be Element of E;
now assume AS: 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 AS,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 a is F-primitive by FIELD_7:8;
  end;
hence thesis by FIELD_7:5;
end;
