Remarks

• In a previous proposition, we saw that, for any power r = p^b of p, the subset {x F | x^r = x} is a subfield of F. Apparently,
  • for r = q, the subfield coincides with F;
  • the subfield only depends on the value of b mod a, where q = p^a.

• Note that x^{q - 1} = 1 for nonzero x in F.