3 The relative class number
The inclusion \(\mathcal O_{K^+}\subseteq \mathcal O_K\) induces the class-group homomorphism
For an odd prime cyclotomic field, the map
is injective.
It suffices to show that the kernel is trivial. If an ideal class maps to the trivial class, then the corresponding extended ideal is principal. By theorem 2.10, the original ideal is principal, so the original class is trivial.
The plus class number divides the full class number:
The injective class-group map identifies \(\mathrm{Cl}(()\mathcal O_{K^+})\) with a subgroup of \(\mathrm{Cl}(()\mathcal O_K)\). Lagrange’s theorem for finite groups gives the divisibility of the corresponding cardinalities.
The relative class number is the natural-number quotient
The class number factors as
This is the definition of \(h^-\) together with theorem 3.3.
Assume that a separate argument proves
Then
Rewrite \(h(K)\) as \(h^+(K)h^-(K)\) using theorem 3.5. If \(p\mid h(K)\), primality of \(p\) gives \(p\mid h^+(K)\) or \(p\mid h^-(K)\). The first alternative is converted to \(p\mid h^-(K)\) by the supplied implication. The converse is immediate because \(h^-(K)\) is a factor of \(h(K)\).