Ideal class group  From Wikipedia, the free encyclopedia
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navigation, search  In mathematics, 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 a certain group known as an 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.  History and origin of the ideal class group[edit]
Ideal class groups (or, rather, what were effectively ideal
class groups) were studied some time before the idea of an ideal
was formulated. These groups appeared in the theory of quadratic
forms: in the case of binary integral q