Tohoku Mathematical Journal

Twisted Kummer and Kummer-Artin-Schreier theories

Noriyuki Suwa

Full-text: Open access

Abstract

We discuss an analogue of the Kummer and Kummer-Artin-Schreier theories, twisting by a quadratic extension. The argument is developed not only over a field but also over a ring, as generally as possible.

Article information

Source
Tohoku Math. J. (2), Volume 60, Number 2 (2008), 183-218.

Dates
First available in Project Euclid: 7 July 2008

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1215442871

Digital Object Identifier
doi:10.2748/tmj/1215442871

Mathematical Reviews number (MathSciNet)
MR2428860

Zentralblatt MATH identifier
1145.13005

Subjects
Primary: 13B05: Galois theory
Secondary: 14L15: Group schemes 12G05: Galois cohomology [See also 14F22, 16Hxx, 16K50]

Citation

Suwa, Noriyuki. Twisted Kummer and Kummer-Artin-Schreier theories. Tohoku Math. J. (2) 60 (2008), no. 2, 183--218. doi:10.2748/tmj/1215442871. https://projecteuclid.org/euclid.tmj/1215442871


Export citation

References

  • M. Demazure and P. Gabriel, Groupes algébriques, Tome I, Masson & Cie, Editeur, Paris; North-Holland Publishing, Amsterdam, 1970.
  • P. Furtwängler, Über die Reziprozitätsgesetze der $l$-ten Potenzreste in algebraischen Zahlkörpern, wenn $l$ eine ungerade Primzahl bedeutet, Math. Ann. 58 (1904), 1--50.
  • P. Furtwängler, Allgemeiner Existenzbeweis fÜr den Klassenkörper eines bebliegen Zahlkörpers, Math. Ann. 63 (1907), 1--37.
  • A. Grothendieck, Le groupe de Brauer, Dix exposés sur la cohomologie des schémas, North-Holland (1968), 46--188.
  • M. Kida, Kummer theory for norm algebraic tori, J. Algebra 293 (2005), 427--447.
  • T. Komatsu, Arithmetic of Rikuna's generic cyclic polynomial and generalization of Kummer theory, Manuscripta Math. 114 (2004), 265--279.
  • Y. Rikuna, On simple families of cyclic polynomials, Proc. Amer. Math. Soc. 130 (2002), 33--35.
  • T. Sekiguchi and N. Suwa, Théories de Kummer-Artin-Schreier, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 417--420.
  • T. Sekiguchi and N. Suwa, On the structure of the group scheme $\boldsymbol Z[\boldsymbol Z/p^n]^\times$, Compos. Math. 97 (1995), 253--271.
  • T. Sekiguchi and N. Suwa, Théorie de Kummer-Artin-Schreier et applications, J. Théor. Nombres Bordeaux 7 (1995), 177--189.
  • T. Sekiguchi, F. Oort and N. Suwa, On the deformation of Artin-Schreier to Kummer, Ann. Sci. école Norm. Sup. (4) 22 (1989), 345--375.
  • J. P. Serre, Groupes algébriques et corps de classes, Hermann, Paris, 1959.
  • W. C. Waterhouse, Introduction to affine group schemes, Springer, 1979.
  • W. C. Waterhouse, A unified Kummer-Artin-Schreier sequence, Math. Ann. 277 (1987), 447--451.
  • W. C. Waterhouse and B. Weisfeiler, One-dimensional affine group schemes, J. Algebra 66 (1980), 550--568.