Note on the class-number of the maximal real subfield of a cyclotomic field (Q1081637)

Let H(m) be the class number of the real cyclotomic field \(K={\mathbb{Q}}(\zeta _ m+\zeta _ m^{-1})\), where \(\zeta _ m\) is a primitive root of unity. Let \(h^ +(m)\) be the class number in the narrow sense of the real quadratic field \(k={\mathbb{Q}}(\sqrt{m})\). This paper gives a relation between \(h^ +(m)\) and H(m). \textit{N. C. Ankeny}, \textit{S. Chowla} and \textit{H. Hasse} [J. Reine Angew. Math. 217, 217-220 (1965; Zbl 0128.035)] and \textit{S.-D. Lang} [ibid. 290, 70-72 (1977; Zbl 0346.12003)] gave some conditions of a prime p satisfying \(H(p)>1\). This paper extends the results of Ankeny, Chowla, Hasse and Lang. Especially, this paper constructs an infinite numerical series of integers m satisfying \(3| H(m)\). This paper also shows that there exist infinitely many square-free integers m satisfying \(n| H(m)\) for any given natural number n.