Around Kummer-Artin-Screier-Witt theories

给定域K和有限群G，用Galois群G构造K的Galois扩张是一个经典问题。最重要的结果是Kummer理论，它断言，如果一个正整数n在K中是可逆的，并且所有的n次单位根都包含在K中，则K的所有n次循环扩张都是通过邻接方程t^n=a的根得到的。0，Artin-Schreier理论认为K的所有p次循环扩张都是由方程t^p-t=a的根得到的。Witt向量理论给出了特征p&gt；0域的p^n次循环扩张的一个很好的刻画。如今，在Galois上同调的框架下证明Kummer、Artin-Schreier和Artin-Schreier-Witt理论是标准的。例如，库默理论源于被称为库默序列的群方案的精确序列和被称为希尔伯特9…的伽罗华上同调的消失定理更多的0。Sekiguchi和Suwa构造了群方案的精确序列，它们统一了Kummer序列和Artin-Schreier-Witt序列。最近，另一个问题引起了专家们的兴趣，他们从Kummer理论中删除了K包含所有n次单位根的条件。小松建立了Kummer理论的一个变种，通过二次扩展扭曲了Kummer理论。在这个研究项目中，Suwa推广了环上的扭曲Kummer理论，澄清了小松的扭曲Kummer理论和Rikna的循环扩张的一般多项式理论之间的关系。此外，Suwa建立了扭曲Kummer理论和Artin-Schreier理论的统一理论，在这项工作中，环的二次扩张的酉群方案起着重要的作用。利用二次扩张的正则表示，我们还得到了扭转Kummer理论和扭转Kummer-Artin-Schreier理论紧化的一个很好的刻画。[1]Rikuna型一般循环多项式的T.Komatsu算法和Kummer理论的推广。马努斯克里塔数学114(2004)265-279[2]Y·里库纳--关于循环多项式的单族。程序阿默。数学课。SoC。130(2002)2215-2218

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