A note on the field isomorphism problem of X^3+sX+s and related cubic Thue equations

TL;DR

This paper investigates when the cubic polynomial X^3+sX+s is isomorphic to related cubic fields by analyzing primitive solutions to specific cubic Thue equations, providing insights into the field isomorphism problem for these polynomials.

Contribution

It introduces a novel approach linking the field isomorphism problem of certain cubic polynomials to solutions of specialized cubic Thue equations.

Findings

Characterization of when X^3+sX+s is isomorphic to related cubic fields.

Explicit conditions on parameters for field isomorphism.

Connection between primitive solutions of Thue equations and field isomorphism.

Abstract

We study the field isomorphism problem of cubic generic polynomial $X^3+sX+s$ over the field of rational numbers with the specialization of the parameter $s$ to nonzero rational integers $m$ via primitive solutions to the family of cubic Thue equations $x^3-2mx^2y-9mxy^2-m(2m+27)y^3=\lambda$ where $\lambda^2$ is a divisor of $m^3(4m+27)^5$.