On cubic Thue equations and the common index divisors of cyclic cubic fields

TL;DR

This paper studies the common index divisors in cyclic cubic fields by solving specific Thue cubic equations under certain parity and divisibility conditions, advancing understanding of algebraic number theory structures.

Contribution

It provides new solutions to Thue cubic equations related to cyclic cubic fields and explores conditions affecting their common index divisors.

Findings

Identifies conditions under which solutions exist for the given Thue equations.

Establishes relationships between the solutions and the structure of cyclic cubic fields.

Advances the classification of common index divisors in these fields.

Abstract

In this paper, we investigate the common index divisors of cyclic cubic fields. Let $a,b,c,d$ and $k$ are integers, we then solve the following Thue cubic equations:: \[ax^3+bx^2y+cxy^2+dy^3= k\ \] when $a,bc+d$ are odd and $3$ doesn't divide $v_2(k)$.