On an index formula for the relative class number of a cyclotomic number field (Q752073)

Let \(h^-\) denote the relative class number of the cyclotomic field \(\mathbb Q(\zeta)\), where \(\zeta\) is a primitive \(p^n\)-th root of 1 \((p\) an odd prime, \(n\ge 1)\). Set \(\eta =p(1+\zeta)/(1-\zeta)\). \textit{K. Girstmair} [ibid. 32, No.1, 100--110 (1989; Zbl 0675.12002)] established an index formula, valid for any imaginary abelian field, which in this case reads \[ ((1-J)\mathbb Z[\zeta] : \mathbb Z[G]\eta)=ch^-. \tag{*} \] Here \(J\) stands for the complex conjugation, \(G= \mathrm{Gal}(\mathbb Q(\zeta)/\mathbb Q)\), and \(c\) is an explicitly given rational integer. The author deduces (*) from the class number formula \(p^{n(r- 1)}h^-=| D|\), where \(r=(p-1)p^{n-1}/2\) and \(D\) denotes the so-called Maillet determinant. The latter formula was proved for \(n=1\) by \textit{L. Carlitz} and \textit{F. R. Olson} [Proc. Am. Math. Soc. 6, 265--269 (1955; Zbl 0065.02703)] and in the general case by the reviewer [Ann. Univ. Turku., Ser. A I 105, 15 p. (1967; Zbl 0171.01202)].