Note on the relative class number of the \(7^n\)th cyclotomic field (Q2846881)

For an integer \(m\), \(\zeta_m\) denotes a \(m\)th primitive root of unity. Let \(p\) be an odd prime. For an integer \(n\geq 0\), let \(h_n^-\) be the relative class number of the \(p^{n+1}\)th cyclotomic field \(\mathbb Q(\zeta_{p^{n+1}})\). Let \(\mathbb B_n/\mathbb Q\) be the cyclic extension of degree \(p^n\) and conductor \(p^{n+1}\) and \(h_n^*\) be the relative class number of the subfield \( \mathbb B_n(\sqrt{-p})\) of \(\mathbb Q(\zeta_{p^{n+1}})\).NEWLINENEWLINEWhen \(p=3\) (respectively \(5\)), it has been proved by \textit{K. Horie} [J. Lond. Math. Soc., II. Ser. 66, No. 2, 257--275 (2002; Zbl 1011.11072)] that \(\ell \nmid h_n^-\) for any \(n\) if \(\ell^2\not\equiv 1 \bmod 9\) (respectively by \textit{L. C. Washington} [Proc. Am. Math. Soc. 61, 205--208 (1977; Zbl 0349.12003)] that \(\ell^8\not\equiv 1\bmod 100\)). The purpose of this note is to give a generalization for the case \(p=7\):NEWLINENEWLINETheorem. Let \(p=7\) and \(\ell\) be a prime with \(\ell\neq 7\). Then \(\ell \) does not divide \(h_n^-\) for any \(n\geq 0\) if \(\ell^6\not\equiv 1\mod 7^2\) and \(\ell \not\equiv \pm 1\bmod 7\).NEWLINENEWLINETheorem. Let \(p=7\) and \(\ell\) be a prime with \(\ell\not=7\). Then \(\ell\) does not divide \(h_n^*\) for any \(n\geq 0\) if \(\ell^6\not\equiv 1\bmod 7^2\).NEWLINENEWLINEThe proofs start from the analytic class number formulas involving the generalized Bernoulli numbers for \(h_n^*/h_{n-1}^*\) and \(h_n^{-}/h_{n-1}^{-}\) as given in [\textit{L. C. Washington}, Introduction to cyclotomic fields. 2nd ed. Graduate Texts in Mathematics. 83. New York, NY: Springer (1997; Zbl 0966.11047)].NEWLINENEWLINEThe author has also investigated the case \(p=11\). He shows using the work of \textit{H. Ichimura} and \textit{S. Nakajima} [Proc. Japan Acad., Ser. A 88, No. 1, 16--20 (2012; Zbl 1333.11103); J. Math. Soc. Japan 64, No. 1, 317--342 (2012; Zbl 1247.11136)] that \(\ell\nmid h_n^*\) for any \(n\) if \(\ell \) is a primitive root \(\bmod 11^2\). Unfortunately, the corresponding assertion for \(h_n^-\) seems too hard to be established by their method, due to the number of cases to investigate.