In algebraic number theory, a mathematical subdivision, the extent to which unique factorization fails in the ring of integers of an algebraic number body (or more generally a Dedekind domain) can be described by a particular group, known as the ideal class group (or class group). If this group ends, as is the case for the ring of integers of a number body, the order of this group is called the class number.
The multiplicative theory of a Dedekind domain is closely intertwined with the structure of her class group. The class of a Dedekind domain is then and only trivial if the ring is a unique factorization domain. Also see
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