Class number one problem for pure cubic fields of Rudman-Stender type (Q676759)

For integers \(d>0\), \(m>0\), \(r(\neq 0, \pm 1)\) such that \(d=m^3+r\) is cube-free, the field \(\mathbb{Q}(\root 3\of d)\) is called a pure cubic field of Rudman-Stender type if \(3m^2\equiv 0 \pmod r\). In this paper, the author intends to determine all pure cubic fields of Rudman-Stender type with class number one. For this purpose, he uses \textit{S. Louboutin's} lower bound for class numbers of pure cubic number fields in terms of the regulator [cf. Nagoya Math. J. 138, 199-208 (1995; Zbl 0826.11051)], and also the explicit form of the fundamental unit of pure cubic fields of Rudman-Stender type [cf. \textit{R. J. Rudman}, Pac. J. Math. 46, 253-256 (1973; Zbl 0258.12002)]. Combining these two results, the author obtains a new lower bound without regulator for class numbers of pure cubic number fields of Rudman-Stender type, from which he derives exactly five pure cubic fields of Rudman-Stender type of class number one, i.e. \(d=2,5,6,10\) and 12.