Monogenic fields arising from trinomials

Abstract

We call a polynomial monogenic if a root $𝜃$ has the property that $\mathbb{Z}[𝜃]$ is the full ring of integers of $\mathbb{Q}(𝜃)$. Consider the two families of trinomials $x^n+ax+b$ and $x^n+c{x}^{n-1}+d$. 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.