The class number h of a cyclotomic field factors as \(h=h^+h^-\), where \(h^+\) is the class number of the maximal real subfield and \(h^-\) is an integer called the relative class number. The author shows that there are only finitely many cyclotomic fields K such that the odd part of \(h^-_ K\) is below a given bound. The proof uses a nice combination of work of \textit{K. Uchida} [Tôhoku Math. J., II. Ser. 23, 97-104 (1971; Zbl 0213.069)] and \textit{E. C. Friedman} [Invent. Math. 65, 425-440 (1982; Zbl 0495.12007)]. It is also shown that \(K={\mathbb{Q}}(\zeta_ m)\) has \(h^-_ K=2^ r\) with \(r\geq 1\) if and only if \(m\in \{29,39,56,65,68,120\}\).