The Hasse norm principle for the maximal real subfields of cyclotomic fields (Q1903387)

Let \(K\) be an algebraic number field of finite degree. It is said that the Hasse norm principle (abbreviated to HNP) holds for \(K\) if \(\mathbb{Q}\cap N_{K/\mathbb{Q}} J_K= N_{K/\mathbb{Q}} K^\times\), where \(J_K\) denotes the idele group of \(K\). \textit{S. Gurak} [J. Reine Angew. Math. 299, 16-27 (1978; Zbl 0367.12006)] and \textit{F. Gerth} [J. Reine Angew. Math. 303, 249-252 (1978; Zbl 0384.12005)] gave necessary and sufficient conditions for HNP to hold for the \(m\)th cyclotomic field \(\mathbb{Q}(\zeta_m)\) with \(m\not\equiv 2\pmod 4\). These results are proved using the fact that HNP holds for an elementary Abelian \(\ell\)-extension \(K/\mathbb{Q}\) with \([K: \mathbb{Q}]= \ell^r\), where \(\ell\) is a prime, if and only if the rank of a certain matrix is \({1\over 2}r(r- 1)\). In this paper, using a similar method the author gives necessary and sufficient conditions for HNP to hold for the maximal real subfield \(\mathbb{Q}(\zeta_m)^+\) of \(\mathbb{Q}(\zeta_m)\). As numerical results he also gives lists of \(m\) with \(m\leq 1200\) such that HNP fails to hold for \(\mathbb{Q}(\zeta_m)\) and for \(\mathbb{Q}(\zeta_m)^+\).