Gradings of Galois extensions (Q6608052)

This paper deals with the gradings of finite field extensions in which all homogeneous components are one-dimensional. Such gradings are called \textit{fine}.\N\NKummer extensions, obtained by adjoining roots of elements from the base field, admit a natural grading based on the Galois group, and all homogeneous components are one-dimensional in this case. In cogalois theory, developed by Greither and Harrison, which is dual to Galois theory, a class of algebraic extensions, Kneser extensions, that admit a natural grading, exists.\N\NIn [\textit{D. A. Badulin} and \textit{A. L. Kanunnikov}, Mosc. Univ. Math. Bull. 77, No. 2, 97--101 (2022; Zbl 1504.16074); translation from Vestn. Mosk. Univ., Ser. I 77, No. 2, 67--71 (2022)], the authors found all gradings of quadratic Kummer extensions. In this paper, they describe all possible gradings of any Kummer extensions, investigate a set of fine gradings for Galois extensions, and show some applications to computing Galois groups.