Around Kummer theories, from the view point of group schemes

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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.