Ideal class group

This article defines a natural context where a group occurs, or is associated, with another algebraic, topological, analytic or discrete structure
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This term is related to: algebraic number theory
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Definition

Let $K$ be a number field (viz an algebraic extension of $\mathbb {Q}$ of finite degree). Let $O$ be the ring of integers in $K$ (viz the elements of $K$ that satisfy monic polynomials with integer coefficients). Then, the ideal class group can be defined in the following steps:

• We can consider the set of all fractional ideals on $O$. A fractional ideal is a subset $J$ of $K$ such that there exists $x\in O$ for which $xJ$ is an ordinary ideal in $O$.
• We can define a multiplication on fractional ideals $I$ and $J$. The product of two fractional ideals is the ideal generated by all products of elements from the two ideals. Under this multiplication, the set of all fractional ideals gets the structure of a monoid.
• The subset of this comprising principal fractional ideals, is a submonoid
• The quotient of the monoid of all fractional ideals by the submonoid of principal fractional ideals, turns out to be a group, and this group is the ideal class group.

The class number is defined to be the order of the ideal class group.

Facts

Realization

See here for more details on what finite abelian groups can be realized as ideal class groups.

Examples

Quadratic number fields

This table lists the ideal class group of quadratic number fields, that is, fields of the form $\mathbb {Q} ({\sqrt {d}})$ for $d\in \mathbb {Z}$, for small $d$.

$d$	Ideal class group of $\mathbb {Q} ({\sqrt {d}})$	Ideal class group of $\mathbb {Q} ({\sqrt {-d}})$
1	trivial group	trivial group
2	trivial group	trivial group
3	trivial group	trivial group
5	trivial group	cyclic group:Z2
6	trivial group	cyclic group:Z2
7	trivial group	trivial group
10	cyclic group:Z2	cyclic group:Z2
11	trivial group	trivial group
13	trivial group	cyclic group:Z2
14	trivial group	cyclic group:Z4
15	cyclic group:Z2	cyclic group:Z2

A theorem proven by Heegner states that $\mathbb {Q} ({\sqrt {-d}})$ for $d>0$ has a trivial ideal class group if and only if $d\in \{1,2,3,7,11,19,43,67,163\}$. For this reason, these numbers as sometimes known as the Heegner numbers.

Results

The ideal class group of a number field is the trivial group if and only if the ring of integers $O$ is a principal ideal domain, equivalently a unique factorisation domain (as it is a Dedekind domain.)