Simplest Cubic Fields

Abstract

Let $Q(\alpha )$ be the simplest cubic field, it is known that $Q(\alpha )$ can be generated by adjoining a root of the irreducible equation $x^3-k{x}^2+(k-3)x+1=0$, where $k$ belongs to $Q$. In this paper we have established a relationship between $\alpha$, ${\alpha }^{\prime }$ and $k,k^{\prime }$ where $\alpha$ is a root of the equation $x^3-k{x}^2+(k-3)x+1=0$ and ${\alpha }^{\prime }$ is a root of the same equation with $k$ replaced by $k^{\prime }$ and $Q(\alpha )=Q({\alpha }^{\prime })$.