On the simplest quartic fields and related Thue equations

TL;DR

This paper explicitly solves the field isomorphism problem for a family of simplest quartic polynomials over various fields, linking solutions of related Thue equations to isomorphism classes of these quartic fields.

Contribution

It provides an explicit method to determine when two such quartic polynomials have the same splitting field and establishes a correspondence between solutions of specific Thue equations and quartic field isomorphism classes.

Findings

Infinitely many values of a yield the same splitting field over an infinite field K.

Finitely many algebraic integers a in a number field K produce the same splitting field.

A correspondence between solutions to quartic Thue equations and simplest quartic fields is established.

Abstract

Let $K$ be a field of char $K\neq 2$. For $a\in K$, we give an explicit answer to the field isomorphism problem of the simplest quartic polynomial $X^4-aX^3-6X^2+aX+1$ over $K$ as the special case of the field intersection problem via multi-resolvent polynomials. From this result, over an infinite field $K$, we see that the polynomial gives the same splitting field over $K$ for infinitely many values $a$ of $K$. We also see by Siegel's theorem for curves of genus zero that only finitely many algebraic integers $a\in\mathcal{O}_K$ in a number field $K$ may give the same splitting field. By applying the result over the field $\mathbb{Q}$ of rational numbers, we establish a correspondence between primitive solutions to the parametric family of quartic Thue equations \[ X^4-mX^3Y-6X^2Y^2+mXY^3+Y^4=c, \] where $m\in\mathbb{Z}$ is a rational integer and $c$ is a divisor of $4(m^2+16)$, and isomorphism classes of the simplest quartic fields.