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We present a generalization of Kummer extensions in algebraic function fields with finite field of constants F q , using the action of Carlitz-Hayes. This generalization of Kummer type extensions is due to Wen-Chen Chi and Anly Li and due also to Fred Schultheis. The main results of this article are Proposition 3.2 and Theorem 3.4. They provide a partial analogue of a theorem of Kummer, which establishes a bijection between Kummer extensions L/K of exponent n and subgroups of K * containing (K *) n .