In this paper, we generalize the results of [M. Sánchez-Mirafuentes and G. Villa–Salvador, Radical extensions for the Carlitz module, J. Algebra398 (2014) 284–302] to rank one Drinfeld modules with class number one. We show that, in the present form, there does not exist a cogalois theory for Drinfeld modules of rank or class number larger than one. We also consider the torsion of the Carlitz module for the extension $𝔽_{q} (T) (Λ_{P^{n}}) / 𝔽_{q} (T) (Λ_{P})$ .

Communicated by J. H. Bruinier

Keywords:

AMSC: 11R60, 11R18, 11R32, 11R58

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