Let \(D\) be a square-free rational integer of the form \(D = 4m^2+1\) or \(m^2+4\) \((m\in \mathbb{N})\). Then, S. Chowla and the reviewer conjectured that there exist exactly 6 real quadratic fields \(\mathbb{Q}(\sqrt{D})\) of class number one in each case [cf. H. Yokoi, Class number and fundamental units of algebraic number fields, Proc. Int. Conf. Katata, Japan 1986, 125–137 (1986; Zbl 0612.12010)]. Afterwards, R. A. Mollin and H. C. Williams [Computational number theory, Proc. Colloq., Debrecen/Hung. 1989, 95–101 (1991; Zbl 0734.11058)] proved this conjecture under the assumption of the generalized Riemann hypothesis.
In this paper, the authors intend to obtain an analog of this result for real quadratic function fields. Namely, for the finite field \(\mathbb{F}_q\) with \(q\) elements put \(k= \mathbb{F}_q(x)\), \(K= k(\sqrt{D(x)})\), where \(D(x)= A(x)^2+ a\) is a square-free polynomial in \(\mathbb{F}_q[x]\) with \(\deg A(x)\geq 1\) and \(a\in \mathbb{F}_q^*\). Then, by using the Weil theorem and Riemann-Roch theorem, they prove that there are precisely six such fields with class number one: \(q=3\), \(D= A^2+1\) with \(A= x^3-x\pm 1\), \(x^2+1\), \(x^2\pm x-1\), \(q=5\), \(D= x^4+2\).