3 The relative class number
The inclusion \(\mathcal O_{K^+}\subseteq \mathcal O_K\) induces, by extension of fractional ideals, the class-group homomorphism
For an odd prime cyclotomic field, the map
is injective.
A group homomorphism is injective when its kernel is trivial. If an ideal class of \(\mathcal O_{K^+}\) maps to the trivial class, a representative ideal becomes principal after extension to \(\mathcal O_K\); by theorem 2.10 the representative was already principal, so the class is trivial.
The plus class number divides the full class number:
By theorem 3.2, \(\mathrm{Cl}(\mathcal O_{K^+})\) embeds as a subgroup of the finite group \(\mathrm{Cl}(\mathcal O_K)\), and Lagrange’s theorem gives the divisibility of cardinalities.
The relative class number is the natural-number quotient
which is well defined by theorem 3.3.
The class number factors as
Multiply the defining quotient of definition 3.4 by \(h^+(K)\); this is legitimate in \(\mathbb {N}\) because \(h^+(K) \mid h(K)\) (theorem 3.3) and \(h^+(K) \ne 0\).
Assume that a separate argument proves the implication
Then
Write \(h(K) = h^+(K)\, h^-(K)\) by theorem 3.5. If \(p \mid h(K)\), then by primality \(p\mid h^+(K)\) or \(p\mid h^-(K)\), and the first alternative implies the second by hypothesis; either way \(p \mid h^-(K)\). Conversely \(h^-(K)\) divides \(h(K)\), so \(p \mid h^-(K)\) implies \(p \mid h(K)\).