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In this paper, we study the representation theory of Hopf-Ore extensions of group algebras and pointed Hopf algebras of rank one over an arbitrary field . Let = (,,) be a Hopf-Ore extension of   and   a rank one quotient Hopf algebra of , where  is a field,  is a group,  is a central element of  and  is a -valued character for  with ()≠1. We first show that the simple weight modules over  and   are finite dimensional. Then we describe the structures of all simple weight modules over  and  , and classify them. We also consider the decomposition of the tensor product of two simple weight modules over   into the direct sum of indecomposable modules. Furthermore, we describe the structures of finite dimensional indecomposable weight modules over  and  , and classify them. Finally, when () is a primitive -th root of unity for some ≥2, we determine all finite dimensional indecomposable projective objects in the category of weight modules over  .

Representations of Hopf-Ore Extensions of Group Algebras and Pointed Hopf Algebras of Rank One