Positive characteristic analogues of finite algebraic numbers

TL;DR

This paper introduces a positive characteristic analogue of finite algebraic numbers over function fields, exploring foundational properties and extending Rosen's concept from number fields to function fields.

Contribution

It defines and studies the properties of the positive characteristic analogue of finite algebraic numbers over rational function fields, expanding the theory into new algebraic settings.

Findings

Established the structure of the ring $\mathcal{P}^0_{\mathcal{A}_K}$

Compared properties with classical finite algebraic numbers

Provided foundational results for future research

Abstract

J.~Rosen introduced the ring $\mathcal{P}^0_{\mathcal{A}}$ of so-called finite algebraic numbers, which may be seen as an analogue of certain periods in the ring $\mathcal{A}=\prod_p \mathbb{Z}/p\mathbb{Z} /\bigoplus_p \mathbb{Z}/p\mathbb{Z}$, $p$ running through all prime numbers. In this article, we introduce its positive characteristic analogue $\mathcal{P}^0_{\mathcal{A}_K}$ over the rational function field $K=\mathbb{F}_q(\theta)$, $q$ being a prime power, and study foundational properties.