Lattice Models of Finite Fields

TL;DR

This paper introduces geometric lattice models of finite fields to make their abstract algebraic properties more accessible and intuitive, especially for undergraduate students, facilitating deeper understanding and further theoretical development.

Contribution

It presents a novel geometric approach to modeling finite fields using lattice structures, bridging abstract algebra and visual intuition.

Findings

Provides concrete lattice models for finite fields

Enhances understanding of Frobenius elements and Artin reciprocity

Accessible to undergraduate students

Abstract

Finite fields form an important chapter in abstract algebra, and mathematics in general. We aim to provide a geometric and intuitive model for finite fields, involving algebraic numbers, in order to make them accessible and interesting to a much larger audience. Such lattice models of finite fields provide a good basis for later developing the theory in a more concrete way, including Frobenius elements, all the way to Artin reciprocity law. Examples are provided, intended for an undergraduate audience in the first place.