Around Kummer-Artin-Screier-Witt theories

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It is a classical problem to construct, given a field K and a finite group G, Galois extensions of K with the Galois group G. The most important result is the Kummer theory, which asserts that, if a positive integer n is invertible in K and all the n-th roots of unity are contained in K, all the cyclic extensions of K of degree n is obtained by adjoining a root of an equation t^n=a. On the other hand, if K is of characteristic p>0, the Artin-Schreier theory asserts that all the cyclic extensions of K of degree p is obtained by adjoining a root of an equation t^p-t=a. The theory of Witt vectors gives an elegant description on cyclic extensions of degree p^n of a field of characteristic p>0.Nowadays it is standard to prove the Kummer, Artin-Schreier and Artin-Schreier-Witt theories in the framework of Galois cohomology. For example, the Kummer theory follows from the exact sequence of group schemes called the Kummer sequence and the vanishing theorem of Galois cohomology called Hilbert 9 … More 0. Sekiguchi and Suwa has constructed exact sequences of group schemes, which unify the Kummer sequences and the Artin-Schreier-Witt sequences.Recently another problem interests specialists to remove from the Kummer theory the condition that K contains all the n-th root of unity. Komatsu established a variant of Kummer theory, twisting the Kummer theory by a quadratic extension. In this research project Suwa generalizes the twisted Kummer theory over a ring, clarifying a relation between the twisted Kummer theory due to Komatsu's and the theory on generic polynomials for cyclic extensions due to Rikuna. Moreover Suwa establishes a theory which unifies the twisted Kummer theory and the Artin-Schreier theory.In this work, the unitary group scheme for a quadratic extension of a ring plays an important role. We have gotten also a nice description on compactifications of the twisted Kummer theory and twisted Kummer-Artin-Schreier theory, using the regular representaion of the quadratic extension.[1] T.Komatsu-Arithmetic of Rikuna's generic cyclic polynomial and generalization of Kummer theory. Manuscripta Math 114(2004) 265-279[2] Y.Rikuna-On simple families of cyclic polynomials. Proc. Amer. Math. Soc. 130 (2002) 2215-2218 Less