Around Kummer theories, from the view point of group schemes

建立和发展类似于伽罗瓦理论的Kummer理论和Artin-Schreier理论，是伽罗瓦理论的重要内容，也是伽罗瓦上同论的出发点，是我们研究的主要目的。不用说，伽罗瓦理论是现代代数的基础。在过去的半个世纪里，上同调理论得到了巨大的发展，它为我们提供了一个强大的算术几何工具。特别地，我们得到了有限平面群格式的几个结果，讨论了裂口Hopf-Galois扩展的正规基与有限平面群格式的单位群格式之间的关系。这是Serre关于伽罗瓦扩展域的正规基与有限群的群代数的单位群之间关系的论证的自然推广。

It is a main purpose of our research to formulate and develop the theories analogous to the Kummer and Artin-Schreier theories, which are important items of the Galois theory and the starting points of the theory of etale cohomology. Needless to say, the Galois theory is a basis of the modern algebra. And the theory of etale cohomology provides us with a powerful tool in arithmetic geometry, which has developed vastly during the past half a century.In particular we obtain several results for finite flat group schemes, concerning to relations between normal bases of a cleft Hopf-Galois extension and the unit group scheme of a finite flat group scheme. This is a natural generalization of Serre's arugument on relations between normal bases of a Galois extension of fields and the unit group of the group algebra of a finite groups.