Editing Fractional ideal (section)

==Number fields==
For the special case of [[Algebraic number field|number fields]] <math>K</math> (such as <math>\mathbb{Q}(\zeta_n)</math>, where <math>\zeta_n</math> = ''exp(2π i/n)'') there is an associated [[ring (mathematics)|ring]] denoted <math>\mathcal{O}_K</math> called the [[ring of integers]] of <math>K</math>. For example, <math>\mathcal{O}_{\mathbb{Q}(\sqrt{d}\,)} = \mathbb{Z}[\sqrt{d}\,]</math> for <math>d</math> [[squarefree integer|square-free]] and [[modular arithmetic|congruent]] to <math>2,3 \text{ }(\text{mod } 4)</math>. The key property of these rings <math>\mathcal{O}_K</math> is they are Dedekind domains. Hence the theory of fractional ideals can be described for the rings of integers of number fields. In fact, [[class field theory]] is the study of such groups of class rings.

=== 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.

=== Structure theorem for fractional ideals ===
One of the important structure theorems for fractional ideals of a [[number field]] states that every fractional ideal <math>I</math> decomposes uniquely up to ordering as
:<math>I = (\mathfrak{p}_1\ldots\mathfrak{p}_n)(\mathfrak{q}_1\ldots\mathfrak{q}_m)^{-1}</math>
for [[prime ideal]]s
:<math>\mathfrak{p}_i,\mathfrak{q}_j \in \text{Spec}(\mathcal{O}_K)</math>.

in the [[spectrum of a ring|spectrum]] of <math>\mathcal{O}_K</math>. For example,
:<math>\frac{2}{5}\mathcal{O}_{\mathbb{Q}(i)}</math> factors as <math>(1+i)(1-i)((1+2i)(1-2i))^{-1} </math>

Also, because fractional ideals over a number field are all finitely generated we can clear denominators by multiplying by some <math>\alpha</math> to get an ideal <math>J</math>. Hence
: <math>I = \frac{1}{\alpha}J</math>
Another useful structure theorem is that integral fractional ideals are generated by up to 2 elements. We call a fractional ideal which is a subset of <math>\mathcal{O}_K</math> ''integral''.