On the Field Isomorphism Problem for the Family of Simplest Quartic Fields

TL;DR

This paper investigates the uniqueness of splitting fields for a family of simplest quartic polynomials, establishing conditions under which two such polynomials share the same field, and linking the problem to elliptic curves.

Contribution

It proves that, under certain conditions, the splitting fields of these polynomials are unique to each polynomial, advancing understanding of the field isomorphism problem for quartic fields.

Findings

At most one other polynomial shares the same splitting field under certain hypotheses.

The proof uses properties of squares in recurrent sequences and a result by J.H.E. Cohn.

Connections to elliptic curves are established in the analysis.

Abstract

Deciding whether or not two polynomials have isomoprhic splitting fields over the rationals is the Field Isomorphism Problem. We consider polynomials of the form $f_n(x) = x^4-nx^3-6x^2+nx+1$ with $n \neq 3$ a positive integer and we let $K_n$ denote the splitting field of $f_n(x)$; a `simplest quartic field'. Our main theorem states that under certain hypotheses there can be at most one positive integer $m \neq n$ such that $K_m=K_n$. The proof relies on the existence of squares in recurrent sequences and a result of J.H.E. Cohn [3]. These sequences allow us to establish uniqueness of the splitting field under additional hypotheses in Section (5) and to establish a connection with elliptic curves in Section (6).