Editing Fractional ideal (section)

=== Associated structures ===
For the ring of integers<ref>{{Cite book|last=Childress|first=Nancy|url=https://www.worldcat.org/oclc/310352143|title=Class field theory|date=2009|publisher=Springer|isbn=978-0-387-72490-4|location=New York|oclc=310352143}}</ref><sup>pg 2</sup> <math>\mathcal{O}_K</math> of a number field, the group of fractional ideals forms a group denoted <math>\mathcal{I}_K</math> and the subgroup of principal fractional ideals is denoted <math>\mathcal{P}_K</math>. The '''[[ideal class group]]''' is the group of fractional ideals modulo the principal fractional ideals, so
: <math>\mathcal{C}_K := \mathcal{I}_K/\mathcal{P}_K</math>
and its class number <math>h_K</math> is the [[order of a group|order]] of the group, <math>h_K = |\mathcal{C}_K|</math>. In some ways, the class number is a measure for how "far" the ring of integers <math>\mathcal{O}_K</math> is from being a [[unique factorization domain]] (UFD). This is because <math>h_K = 1</math> if and only if <math>\mathcal{O}_K</math> is a UFD.

==== Exact sequence for ideal class groups ====
There is an [[exact sequence]]
:<math>0 \to \mathcal{O}_K^* \to K^* \to \mathcal{I}_K \to \mathcal{C}_K \to 0</math>
associated to every number field.