Class number one criterion for some non-normal totally real cubic fields

TL;DR

This paper establishes a criterion for when certain non-normal totally real cubic fields, defined by specific cubic polynomials, have class number one, contributing to the understanding of their algebraic properties.

Contribution

It provides a new class number one criterion specifically for the family of non-normal totally real cubic fields defined by the polynomial $f_m(x)$.

Findings

Derived explicit class number one criterion for the fields

Identified conditions on parameter m for class number one

Enhanced understanding of algebraic properties of these cubic fields

Abstract

Let ${\{K_m\}_{m\geq 4}}$ be the family of non-normal totally real cubic number fields defined by the irreducible cubic polynomial $f_m(x)=x^3-mx^2-(m+1)x-1$, where $m$ is an integer with $m\geq 4$. In this paper, we will give a class number one criterion for $K_m$.