Around KummerArtinScreierWitt theories
Project/Area Number 
16540040

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Chuo University 
Principal Investigator 
SUWA Noriyuki Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (10196925)

CoInvestigator(Kenkyūbuntansha) 
SEKIGUCHI Tsutomu Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (70055234)
MOMOSE Fumiyuki Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (80182187)

Project Period (FY) 
2004 – 2006

Project Status 
Completed (Fiscal Year 2006)

Budget Amount *help 
¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 2006: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2005: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2004: ¥600,000 (Direct Cost: ¥600,000)

Keywords  group scheme / Kummer theory / ArtinSchreier theory / KummerArtinSchreier theory / twisted Kummer theory / fppf cohomology / etale cohomology / 代数群 / equivariant compactification / 生成多項式 
Research Abstract 
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 nth 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 ArtinSchreier theory asserts that all the cyclic extensions of K of degree p is obtained by adjoining a root of an equation t^pt=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, ArtinSchreier and ArtinSchreierWitt 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 ArtinSchreierWitt sequences. Recently another problem interests specialists to remove from the Kummer theory the condition that K contains all the nth 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 ArtinSchreier 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 KummerArtinSchreier theory, using the regular representaion of the quadratic extension. [1] T.KomatsuArithmetic of Rikuna's generic cyclic polynomial and generalization of Kummer theory. Manuscripta Math 114(2004) 265279 [2] Y.RikunaOn simple families of cyclic polynomials. Proc. Amer. Math. Soc. 130 (2002) 22152218 Less

Report
(4 results)
Research Products
(4 results)