Simplest Cubic Fields

TL;DR

This paper explores the algebraic relationships between roots of the simplest cubic fields and their defining parameters, providing insights into their structure and interrelations.

Contribution

It establishes a relationship between roots and parameters of the simplest cubic fields, enhancing understanding of their algebraic structure.

Findings

Derived relationships between roots and parameters $k$ and $k'$

Showed that different parameters can generate the same cubic field

Enhanced understanding of the algebraic structure of 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 $x^{3}-kx^{2}+(k-3)x+1=0$, where $k$ belongs to $Q$. In this paper we have established a relationship between $\alpha$, $\alpha'$ and $k,k'$ where $\alpha$ is a root of the equation $x^{3}-kx^{2}+(k-3)x+1=0$ and $\alpha'$ is a root of the same equation with $k$ replaced by $k'$ and $Q(\alpha)=Q(\alpha')$.