A Carlitz module analogue of the Grunwald--Wang theorem

TL;DR

This paper establishes a Carlitz module analogue of the classical Grunwald--Wang theorem, extending the local-global principle for power elements from number fields to function fields via Carlitz modules.

Contribution

It introduces a new analogue of the Grunwald--Wang theorem within the framework of Carlitz modules, expanding the understanding of local-global principles in function field arithmetic.

Findings

Proves a Carlitz module version of the Grunwald--Wang theorem.

Identifies conditions under which elements are Carlitz module powers locally and globally.

Extends classical number field results to function field setting.

Abstract

The classical Grunwald--Wang theorem is an example of a local--global (or Hasse) principle stating that except in some special cases which are precisely determined, an element $m$ in a number field $\mathbb{K}$ is an $a$-th power in $\mathbb{K}$ if and only if it is an $a$-th power in the completion $\mathbb{K}_{v}$ for all but finitely many primes $v$ of $\mathbb{K}$. In this paper, we prove a Carlitz module analogue of the Grunwald--Wang theorem.