On the unit groups and the ideal class groups of certain cubic number fields

Abstract

Let $f(x)=x^3+3x+a^3(a\in \mathbf{Z})$ be a cubic polynomial and $\theta$ be the real root of $f(x)$. We consider the unit group of $\mathbf{Q}(\theta )$. We show that $\eta =1-a^2-\alpha \theta$ is a fundamental unit of $\mathbf{Q}(\theta )$ under certain conditions. And we consider the 3-class group of $\mathbf{Q}(\theta )$.