Ideal Class Group

The ideal class group provides a way of classifying which Dedekind domains are unique factorisation domains, and in some sense quantifying how far out the domain is from a unique factorisation domain. This is done through the ideal class number.

Let \(\mathbb{K}\) be an algebraic number field and \(\mathcal{O}_\mathbb{K}\) its number ring. Then let \(\mathcal{I}_\mathbb{K}\) be the fractional ideal group of \(\mathcal{O}_\mathbb{K}\), and \(\mathcal{P}_\mathbb{K}\) be the subgroup of principal fractional ideal (this is trivially a subgroup since the inverse is just the principal ideal generated by the inverse, and the product of fractional principal ideals is the ideal generated by the product of the generators). This subgroup is normal because the parent group is abelian.

While this construction is simple in the case of fractional ideals, if we wish to restrict to just normal ideals it requires a bit of additional work. Firstly notice that the cosets of \(\mathcal{P}_\mathbb{K}\) in \(\mathcal{I}_\mathbb{K}\) are equivalence classes of the equivalence relation \(I \sim J \iff I \in J\mathcal{P}_\mathbb{K}\). We will prove that every coset contains at least one normal ideal, and then restrict the equivalence relation to just normal ideals.