In
mathematics, for a field
K an
ideal class group (or
class group) is the quotient group
JK/PK where
JK is the whole
fractional ideals of
K and
PK is the
principal ideals of
K. The extent to which
unique factorization fails in the
ring of integers of an
algebraic number field (or more generally any
Dedekind domain) can be described by the
ideal class group (or
class group). If this group is finite (as it is in the case of the ring of integers of a number field), then the
order of the group is called the
class number. The multiplicative theory of a Dedekind domain is intimately tied to the structure of its class group. For example, the class group of a Dedekind domain is trivial if and only if the ring is a
unique factorization domain.