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arXiv · 1512.09343

Trinomials defining quintic number fields

Jesse Patsolic·Jeremy Rouse
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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.

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Jesse Patsolic, Jeremy Rouse. 2015-12-31. Trinomials defining quintic number fields. https://arxiv.org/abs/1512.09343

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