The quadratic analogue of the simplest cubic fields (Q2716716)

In 1974, \textit{D. Shanks} introduced the simplest cubic fields generated by \(x^3= x^2+ (a+3) x+1\), where \(a^2+ 3a+9\) is prime [cf. Math. Comput. 28, 1137-1152 (1974; Zbl 0307.12005)]. In this paper, the author intends to obtain the quadratic analogue and studies the quadratic fields generated by \(x^2= ax+1\), where \(a^2+4\) is a prime \(p\). In view of Gauss' conjecture on the class-number one problem, it is interesting to find all values of prime \(p\) \((= a^2+4)\) such that the quadratic field \(\mathbb{Q}(\sqrt p)\) has class number 1. NEWLINENEWLINENEWLINEFor this purpose, he first obtains a lower bound of the value of the Dirichlet \(L\)-function \(L(s,\chi_p)\) at \(s=1\) associated to the Legendre symbol \(\chi_p\) modulo \(p\), and by using the Dedekind class number formula he shows that the class number of \(\mathbb{Q}(\sqrt p)\) is equal to ne only for \(p= 5, 13, 29, 53, 173\) possibly with one exception. However, this result is already known in a more general form [cf. \textit{H. Yokoi}, Class number one problem for real quadratic fields, Proc. Japan Acad., Ser. A 64, 53-55 (1988; Zbl 0662.12006)]. Moreover he shows analogous results for the class numbers 3 and 5.