Gradings of Galois Extensions

This paper is devoted to the gradings of finite field extensions in which all homogeneous components are one-dimensional. Such gradings are called fine. Kummer extensions are an important class of extensions that admit fine gradings. There always exists a standard grading of Kummer extension based on the Galois group. The paper describes all fine gradings of Kummer extensions, and, in particular, it establishes a criterion for any fine grading to be isomorphic to the standard one. We also investigate gradings of a wider class of Galois extensions that admit fine gradings.