Monogenic fields arising from trinomials

TL;DR

This paper investigates when certain families of trinomials generate monogenic number fields, providing infinite instances, density results, and specific criteria for monogeneity using advanced algebraic methods.

Contribution

It establishes infinite monogenic cases for two trinomial families and derives necessary and sufficient conditions using the Montes algorithm for specific degrees.

Findings

Families are monogenic infinitely often.

Provides positive density estimates for monogenic cases.

Uses Montes algorithm to characterize monogeneity for degrees 5 and 6.

Abstract

We call a polynomial monogenic if a root $\theta$ has the property that $\mathbb{Z}[\theta]$ is the full ring of integers in $\mathbb{Q}(\theta)$. Consider the two families of trinomials $x^n + ax + b$ and $x^n + cx^{n-1} + d$. For any $n>2$, we show that these families are monogenic infinitely often and give some positive densities in terms of the coefficients. When $n=5$ or 6 and when a certain factor of the discriminant is square-free, we use the Montes algorithm to establish necessary and sufficient conditions for monogeneity, illuminating more general criteria given by Jakhar, Khanduja, and Sangwan using other methods. Along the way we remark on the equivalence of certain aspects of the Montes algorithm and Dedekind's index criterion.