The **ideal class group** of a number field $K$ with ring of integers $O_K$ is the group of equivalence classes of ideals, given by the quotient of the multiplicative group of all fractional ideals of $O_K$ by the subgroup of principal fractional ideals.

Since $K$ is a number field, the ideal class group of $K$ is a finite abelian group, and so has the structure of a product of cyclic groups encoded by a finite list $[a_1,\dots,a_n]$, where the $a_i$ are positive integers with $a_{i+1}\mid a_i$ for $1\le i<n$.

**Knowl status:**

- Review status: reviewed
- Last edited by Holly Swisher on 2019-04-26 14:26:25

**Referred to by:**

- ec.global_minimal_model
- ec.obstruction_class
- nf.class_number
- nf.narrow_class_number
- lmfdb/number_fields/templates/nf-index.html (line 104)
- lmfdb/number_fields/templates/nf-search.html (line 16)
- lmfdb/number_fields/templates/nf-search.html (line 99)
- lmfdb/number_fields/templates/nf-show-field.html (line 73)

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