Positive characteristic analogues of finite algebraic numbers

Summary · qwen2.5:32b

The article introduces $\mathcal{P}^0_{\mathcal{A}_K}$ as a positive characteristic analogue of Rosen's ring of finite algebraic numbers over the rational function field $K=\mathbb{F}_q(\theta)$, foundational properties of which are studied to extend number-theoretic concepts into function fields. This extension is significant for bridging number theory and function field arithmetic. The study provides a new framework for analyzing algebraic structures in positive characteristic settings, exemplified through the examination of properties over $\mathbb{F}_q(\theta)$.

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Exploring the properties of finite algebraic numbers in positive characteristic through the introduction and study of $\mathcal{P}^0_{\mathcal{A}_K}$ over the rational function field $K$. This research could be a stepping stone for future developments in number theory.

Excerpt

arXiv:2601.21209v2 Announce Type: replace 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, and provide further scopes.