With this translation, the classic monograph UEber die Klassenzahl
abelscher Zahlkoerper by Helmut Hasse is now available in English
for the first time. The book addresses three main topics: class
number formulas for abelian number fields; expressions of the class
number of real abelian number fields by the index of the subgroup
generated by cyclotomic units; and the Hasse unit index of
imaginary abelian number fields, the integrality of the relative
class number formula, and the class number parity. Additionally,
the book includes reprints of works by Ken-ichi Yoshino and
Mikihito Hirabayashi, which extend the tables of Hasse unit indices
and the relative class numbers to imaginary abelian number fields
with conductor up to 100. The text provides systematic and
practical methods for deriving class number formulas, determining
the unit index and calculating the class number of abelian number
fields. A wealth of illustrative examples, together with
corrections and remarks on the original work, make this translation
a valuable resource for today's students of and researchers in
number theory.
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