  We produce generalizations of Iwasawa's `Riemann-Hurwitz' formula for number
fields. These generalizations apply to cyclic extensions of number fields of
degree p^n for any positive integer n. We first deduce some congruences and
inequalities and then use these formulas to establish a vanishing criterion for
Iwasawa \lambda-invariants which generalizes a result of Takashi Fukuda et. al.
for totally real number fields.
