Editing Ideal class group (section)

{{Short description|In number theory, measure of non-unique factorization}}
{{more citations needed|date=February 2010}}
In [[mathematics]], the '''ideal class group''' (or '''class group''') of an  [[algebraic number field]] <math>K</math> is the [[quotient group]] <math>J_K/P_K</math> where <math>J_K</math> is the [[group (mathematics)|group]] of [[fractional ideal]]s of the [[ring of integers]] of <math>K</math>, and <math>P_K</math> is its [[subgroup]] of [[principal fractional ideal|principal]] ideals. The class group is a measure of the extent to which [[unique factorization]] fails in the ring of integers of <math>K</math>. The [[order of a group|order]] of the group, which is [[finite group|finite]], is called the '''class number''' of <math>K</math>.

The theory extends to [[Dedekind domain]]s and their [[field of fractions|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 group|trivial]] [[if and only if]] the ring is a [[unique factorization domain]].