This paper is a continuation of a recent work [5] on the imaginary quadratic fields \(\mathbb {Q}(\sqrt{-D_s})\) where \(5D_s = F_{10\,s+5}\) , with \(F_n\) the \(n\) -th Fibonacci number. In a previous paper [5], it was shown that the class number of \(\mathbb {Q}(\sqrt{-D_s})\) is divisible by \(5\) for all nonnegative \(s\) except possibly those \(s \equiv 0 \pmod {20}\) . In this paper, we refine the argument and remove that restriction, proving that the class number of \(\mathbb {Q}(\sqrt{-D_s})\) is divisible by \(5\) for every positive integer \(s\) . Our proof combines a construction of a quintic polynomial, properties of Fibonacci-Lucas numbers, and Sase’s ramification criterion to exhibit an unramified \(C_5\) -extension within a dihedral extension of degree 10.
