Orders in cubic and quartic number fields and classical Diophantine equations

Abstract: An order $\mathcal{O}$ in an algebraic number field is called monogenic if over $\mathbb{Z}$ it can be generated by one element. Győry has shown that there are finitely equivalence classes $\alpha \in \mathcal{O}$ such that $\mathcal{O} = \mathbb{Z}[\alpha]$, where two algebraic integers $\alpha, \alpha'$ are called equivalent if $\alpha + \alpha'$ or $\alpha - \alpha'$ is a rational integer. An interesting problem is to count the number of monogenizations of a given monogenic order. First we will note, for a given order $\mathcal{O}$, that $$\mathcal{O} = \mathbb{Z}[\alpha] \text{ in } \alpha$$ is indeed a Diophantine equation. Then we will discuss how some old algorithmic results can be used to obtain new and improved upper bounds for the number of monogenizations of a cubic or quartic order.

This talk should be accessible to any math graduate student and questions about basic concepts are welcome. We will start by recalling some definitions from elementary algebraic number theory. Number fields, lattices over $\mathbb{Z}$, and simple polynomial equations are the main focus of this talk.

algebraic geometrynumber theory

Audience: researchers in the topic

SFU NT-AG seminar

Series comments: Description: Research/departmental seminar

Seminar usually meets in-person. For online editions, the Zoom link is shared via mailing list.