A note on class numbers of the simplest cubic fields
A simplest cubic field is defined by the irreducible polynomial over the rational number field \(\mathbb{Q}\) of the form \(f(x)= x^3+ mx^2 -(m+3)x +1\) with a rational integer \(m\). For any given rational integer \(n\geq 2\), \textit{K. Uchida} [J. Math. Soc. Japan 26, 447-453 (1974; Zbl 0281.12007)] and \textit{L. C. Washington} [Math. Comput. 48, 371-384 (1987; Zbl 0613.12002)] constructed simplest cubic fields with class numbers divisible by \(n\) under the assumption of \(3 \nmid m\). In this paper, the author intends to weaken the assumption on \(m\), and for \(m \equiv 3\), 21 mod 27, \((m>3)\) he constructs simplest cubic fields with class numbers divisible by \(n\) in a similar way. Namely, let \(a>1\) be an integer satisfying \((a,6) =1\), and \(m>3\) be an integer such that \(m \equiv 3\) or 21 (mod 27) and \(2m +3= 9a^n\). Then, he proves that the class number of the simplest cubic field defined by the irreducible polynomial \(f(x)= x^3+ mx^2- (m+3) x+1\) with such \(m\) is divisible by \(3n\).
scientific article; zbMATH DE number 1037300