Editing Fractional ideal (section)

=== 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''.