A Carlitz module analogue of the Grunwald--Wang theorem

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.