Construction of certain real quadratic fields
In this paper, the author proves the following theorem: For an arbitrarily given natural number n and a square-free rational integer \(k>1\) satisfying some conditions, \(\epsilon =(kn^ 2\pm 2+n\sqrt{m})/2\) with \(m=k(kn^ 2\pm 4)\) is the fundamental unit of the real quadratic field \({\mathbb{Q}}(\sqrt{m})\). This is a partial generalization of the reviewer's former result, which gives the fundamental unit of \({\mathbb{Q}}(\sqrt{m})\) in the same form [J. Number Theory 2, 106-115 (1970; Zbl 0201.057)]. By means of valuation, he also gives another proof of the well-known theorem that there exist infinitely many real quadratic fields with the class number divisible by an arbitrarily given natural number n.
scientific article; zbMATH DE number 3855237