Ideal class group

In mathematics, the ideal class group (or class group) of an algebraic number field $K$ is the quotient group $J K / P K$ where $J K$ is the group of fractional ideals of the ring of integers of $K$ , and $P K$ is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of $K$ . The order of the group, which is finite, is called the class number of $K$ .

The theory extends to Dedekind domains and their fields of fractions, for which the multiplicative properties are intimately tied to the structure of the class group. For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain.