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

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SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

We normally meet in-person in the indicated room. For online editions, we use Zoom and distribute the link through the mailing list. If you wish to be put on the mailing list, please subscribe to ntag-external using lists.sfu.ca

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