  We give a completely normal element in the maximal real subfield of a
cyclotomic field over the field of rational numbers, which is different from
that of Okada. This result is a consequence of the criterion for a normal
element developed in [Normal bases of ray class fields over imaginary quadratic
fields, Math. Zeit.]. Furthermore, we find a completely normal element in
certain extension of modular function fields in terms of a quotient of the
modular discriminant function.
