Letter to the editor (Q5906759)

Let \(K_m=\mathbb{Q}(\zeta_{2^m})\) be the \(2^m\)th cyclotomic field, and let \(h^-(K_m)\) denote its relative class number. The author calls attention to a seemingly forgotten estimate of \textit{S. Hyyrö} [Ann. Acad. Sci. Fenn., Ser. A I 398 (1967; Zbl 0314.12007)] to the effect that \[ h^-(K_m)\leq 2^m(2^{m-2}/16)^{2^{m-3}}. \] A recent paper of \textit{H. Tsumura} [J. Number Theory 60, 70-79 (1996; Zbl 0866.11063)] contains a somewhat weaker result. Without referring to Hyyrö's paper, \textit{K. Feng} [Sci. Sin., Ser. A 27, 1121-1128 (1984; Zbl 0562.12007)] had also derived a complicated formula for \(h^-(K_m)\) which leads to Hyyrö's estimate.