Trinomials defining quintic number fields

TL;DR

This paper explores the relationship between quintic number fields and specific irreducible trinomials, using algebraic curves and elliptic curve techniques to identify all such trinomials in certain cases.

Contribution

It establishes a correspondence between irreducible trinomials and rational points on a genus four curve associated with each quintic field, and applies elliptic curve Chabauty to find these points.

Findings

Constructed genus four curves for quintic fields

Mapped these curves to elliptic curves over number fields

Determined rational points on the curves in some cases

Abstract

Given a quintic number field $K/\mathbb{Q}$, we study the set of irreducible trinomials, polynomials of the form $x^{5} + ax + b$, that have a root in $K$. We show that there is a genus four curve $C_{K}$ whose rational points are in bijection with such trinomials. This curve $C_{K}$ maps to an elliptic curve defined over a number field, and using this map, we are able (in some cases) to determine all the rational points on $C_{K}$ using elliptic curve Chabauty.