We present a generalization of Kummer extensions in algebraic function fields with finite field of constants Fq, using the action of CarlitzHayes. This generalization of Kummer type extensions is due to WenChen 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 .