Class numbers of pure quintic fields
Suppose that \(K=\mathbb{Q}(m^{1/5})\) where \(m\) is a fifth power free integer. In the paper under review, the author proves the following nice result: If a prime \(p\equiv 4 \mod 5\) divides \(m\), then \(5\) divides the class number of \(K\). This is a particular case of a general conjecture of the author for the class number of the algebraic number field \(\mathbb Q(m^{1/\ell})\). To achieve his goal for the number field \(K\), the author obtains new results on the arithmetic (class number, unit group) of the maximal real subfield \(L^+\) of the Galois closure \(L\) of \(K\). The introduction in the paper starts with an interesting survey of previous results of \textit{T. Honda} [J. Number Theory 3, 7--12 (1971; Zbl 0222.12004)], \textit{C. J. Parry} and \textit{C. D. Walter} [Mathematika 24, 133 (1977; Zbl 0364.12006); ibid. 23, 220--226 (1976; Zbl 0364.12005)], and \textit{K. Iimura} [J. Reine Angew. Math. 292, 201--210 (1977; Zbl 0351.12003)] on the divisibility of the class number of an algebraic number field by a prime number.
scientific article; zbMATH DE number 6516879