2 Class numbers of the cyclotomic CM field

In this chapter \(K\) denotes the \(p\)th cyclotomic field for an odd prime \(p\). The goal is the factorisation \(h = h^+ h^-\) of the class number: we prove that the natural map on class groups induced by the inclusion \(\mathcal O_{K^+} \subseteq \mathcal O_K\) is injective, so that \(h^+ \mid h\) and the relative class number \(h^- = h/h^+\) makes sense as a natural number. The key step is a descent statement for principal ideals (theorem 2.10): an ideal of \(\mathcal O_{K^+}\) that becomes principal in \(\mathcal O_K\) was principal to begin with.