Editing Algebraic number theory (section)

===Ideal class group===
Unique factorization fails if and only if there are prime ideals that fail to be principal. The object which measures the failure of prime ideals to be principal is called the ideal class group. Defining the ideal class group requires enlarging the set of ideals in a ring of algebraic integers so that they admit a [[group (mathematics)|group]] structure. This is done by generalizing ideals to [[fractional ideal]]s. A fractional ideal is an additive subgroup {{math|''J''}} of {{math|''K''}} which is closed under multiplication by elements of {{math|''O''}}, meaning that {{math|''xJ'' ⊆ ''J''}} if {{math|''x'' ∈ ''O''}}. All ideals of {{math|''O''}} are also fractional ideals. If {{math|''I''}} and {{math|''J''}} are fractional ideals, then the set {{math|''IJ''}} of all products of an element in {{math|''I''}} and an element in {{math|''J''}} is also a fractional ideal. This operation makes the set of non-zero fractional ideals into a group. The group identity is the ideal {{math|1=(1) = ''O''}}, and the inverse of {{math|''J''}} is a (generalized) [[ideal quotient]]:

:<math>J^{-1} = (O:J) = \{x \in K: xJ \subseteq O\}.</math>

The principal fractional ideals, meaning the ones of the form {{math|''Ox''}} where {{math|''x'' ∈ ''K''<sup>×</sup>}}, form a subgroup of the group of all non-zero fractional ideals. The quotient of the group of non-zero fractional ideals by this subgroup is the ideal class group. Two fractional ideals {{math|''I''}} and {{math|''J''}} represent the same element of the ideal class group if and only if there exists an element {{math|''x'' ∈ ''K''}} such that {{math|1=''xI'' = ''J''}}. Therefore, the ideal class group makes two fractional ideals equivalent if one is as close to being principal as the other is. The ideal class group is generally denoted {{math|Cl ''K''}}, {{math|Cl ''O''}}, or {{math|Pic ''O''}} (with the last notation identifying it with the [[Picard group]] in algebraic geometry).

The number of elements in the class group is called the '''class number''' of ''K''. The class number of {{math|'''Q'''(√{{Overline|-5}})}} is 2. This means that there are only two ideal classes, the class of principal fractional ideals, and the class of a non-principal fractional ideal such as {{math|(2, 1 + √{{Overline|-5}})}}.

The ideal class group has another description in terms of [[divisor (algebraic geometry)|divisor]]s. These are formal objects which represent possible factorizations of numbers. The divisor group {{math|Div ''K''}} is defined to be the [[free abelian group]] generated by the prime ideals of {{math|''O''}}. There is a [[group homomorphism]] from {{math|''K''<sup>×</sup>}}, the non-zero elements of {{math|''K''}} up to multiplication, to {{math|Div ''K''}}. Suppose that {{math|''x'' ∈ ''K''}} satisfies
:<math>(x) = \mathfrak{p}_1^{e_1} \cdots \mathfrak{p}_t^{e_t}.</math>
Then {{math|div ''x''}} is defined to be the divisor
:<math>\operatorname{div} x = \sum_{i=1}^t e_i[\mathfrak{p}_i].</math>
The [[kernel (algebra)|kernel]] of {{math|div}} is the group of units in {{math|''O''}}, while the [[cokernel]] is the ideal class group. In the language of [[homological algebra]], this says that there is an [[exact sequence]] of abelian groups (written multiplicatively),
:<math>1 \to O^\times \to K^\times \xrightarrow{\text{div}} \operatorname{Div} K \to \operatorname{Cl} K \to 1.</math>