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Study of the structure of integer rings from the view point of cyclotomic Iwasawa theory

Research Project

Project/Area Number 16540033
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionIbaraki University (2005-2006)
Yokohama City University (2004)

Principal Investigator

ICHIMURA Humio  Ibaraki University, College of Science, Professor, 理学部, 教授 (00203109)

Co-Investigator(Kenkyū-buntansha) NAITO Hirotada  Kagawa University, Faculty of Education, Professor, 教育学部, 教授 (00180224)
KOYA Yoshihiro  Yokohama City University, International College of Arts and Science, Associate Professor, 国際総合科学部, 準教授 (50254230)
SUMIDA Hiroki  Tokushima University, Graduate School of Enginnering, Associate Professor, 大学院ソシオテクノサイエンス研究部, 助教授 (90291476)
AIBA Akira  Ibaraki University, College of Science, Associate Professor, 理学部, 助教授 (90202457)
Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2006: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2005: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2004: ¥700,000 (Direct Cost: ¥700,000)
Keywordsnormal integral basis / ideal class group / Stickelberger ideal / p-整数環 / 単項化定理 / descent / Kummer拡大 / Kummer duality
Research Abstract

We studied on the Hilbert-Speiser number fields and Stickelberger ideals. A number field F satisfies the Hilbert-Speiser condition (H_p) at a prime p when any tame cyclic extension N/F of degree p has a normal integral basis. By Hilbert and Speiser, the rationals Q satisfy the condition for all primes p. on the other hand, it is shown by Greither et al. that any F different from Q does not satisfy (H_p) for infinitely many p. Thus, it becomes of interest to ask which (F, p) satisfies the condition. As we can see in Iwasawa theory, the ring of p-integers is an object more natural than the usual integer ring when we deal with p-extensions. We denote by (H_p') the corresponding condition for the p-integer ring. We obtained many results on these conditions. Most impreesiveresult is a relation between them and Stickelberger ideals. Let G be the multiplicative group of the finite field of order p, and let S be the classical Stickelberger ideal of the group ring Z[G]. For each subgroup H of G, we define an ideal S_H of Z[H] as a H-part of S. Let K=F(ζ_p) and H=Gal(K/F). Regarding H as a subgroup of G, the ideal S_H can act on the p-ideal class group Cl_K of K. Most important results are (1) that F satisfies (H_p') if and only if S_H kills Cl_K, (2) that we obtained several properties of the ideal S_H, and as an application (3) that we "determined" the subfields of the p-cyclotomic field satisfying (H_p').

Report

(4 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (36 results)

All 2007 2006 2005 2004

All Journal Article (36 results)

  • [Journal Article] Computation of the p-part of the ideal class group of certain real abelian fields2007

    • Author(s)
      隅田 浩樹
    • Journal Title

      Mathematics of Computation 76

      Pages: 1059-1071

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Computation of the p-part of the ideal class group of certain real abelian fields2007

    • Author(s)
      Sumida, Hiroki
    • Journal Title

      Mathematics of Computation vol.76

      Pages: 1056-1071

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Stickelberger ideals of conductor p and their application2006

    • Author(s)
      市村 文男
    • Journal Title

      Journal of the Mathematical Society of Japan 58

      Pages: 885-902

    • NAID

      10018381175

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Normal integral bases and ray class groups, II2006

    • Author(s)
      市村 文男
    • Journal Title

      Yokohama Mathematical Journal 53

      Pages: 75-81

    • NAID

      120001740834

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Triviality of Stickelberger ideals of conductor p2006

    • Author(s)
      市村 文男
    • Journal Title

      Journal of Mathematical Sciences, the University of Tokyo 13

      Pages: 617-628

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Stickelberger ideals and normal bases of rings of p-integers2006

    • Author(s)
      市村 文男
    • Journal Title

      Mathematical Journal of Okayama University 48

      Pages: 9-20

    • NAID

      120002309514

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] A class number formula for the p-cyclotomic field2006

    • Author(s)
      市村 文男
    • Journal Title

      Archiv der Mathematik 87

      Pages: 539-545

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Note on the rings of integers of certain tame 2-Galois extensions over a number field2006

    • Author(s)
      市村 文男
    • Journal Title

      Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg 76

      Pages: 63-68

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Stickelberger ideals of conductor and their application2006

    • Author(s)
      Ichimura, Humio
    • Journal Title

      Journal of the Mathematical Society of Japan vol.58

      Pages: 885-902

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Normal integral bases and ray class groups, II2006

    • Author(s)
      Ichimura, Humio
    • Journal Title

      Yokohama Mathematical Journal vol.53

      Pages: 75-81

    • NAID

      120001740834

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Triviality of Stickelberger ideals of conductor p2006

    • Author(s)
      Ichimura, Humio
    • Journal Title

      Journal of Mathematical Sciences, the University of Tokyo vol 13

      Pages: 617-628

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Stickelberger ideals and normal bases of rings of p-integers2006

    • Author(s)
      Ichimura, Humio
    • Journal Title

      Mathematical Journal of Okayama University vol.48

      Pages: 9-20

    • NAID

      120002309514

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A class number formula for the p-cyclotomic field2006

    • Author(s)
      Ichimura, Humio
    • Journal Title

      Archiv der Mathematik vol.87

      Pages: 539-545

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Note on the rings of integers of certain tame 2-Galois extensions over a number field2006

    • Author(s)
      Ichimura, Humio
    • Journal Title

      Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg vol.76

      Pages: 63-68

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Stickelberger ideals of conductor p and their application2006

    • Author(s)
      市村 文男, 隅田 浩樹
    • Journal Title

      Journal of the Mathematical Society of Japan 58

      Pages: 885-902

    • NAID

      10018381175

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Note on the rings of integers of certain tame-2-Galois extensions over a number field2006

    • Author(s)
      市村 文男
    • Journal Title

      Abhandlungen aus dem Mathematischen Seminar der Universitat de Hamburg 76

      Pages: 63-68

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On a theorem of Kawamoto on normal bases of rings of integers, II2005

    • Author(s)
      市村 文男
    • Journal Title

      Canadian Mathematics Bulletine 48

      Pages: 576-579

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the ring of p-integers of a cyclic p-extension over a number field2005

    • Author(s)
      市村 文男
    • Journal Title

      Journal de Theorie des Nombres Bordeaux 17

      Pages: 779-786

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The Iwasawa invariants and the higher K-groups associated to real quadratic fields2005

    • Author(s)
      隅田 浩樹
    • Journal Title

      Experimental Mathematics 14

      Pages: 307-316

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On a theorem of Kawamoto on normal bases of rings of integers, II2005

    • Author(s)
      Ichimura, Humid
    • Journal Title

      Canadian Mathematical Bulletin vol.48

      Pages: 576-579

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the ring of p-integers of a cyclic p-extension over a number field2005

    • Author(s)
      Ichimura, Humio
    • Journal Title

      Journal de Theorie des Nombres Bordeaux vol.17

      Pages: 779-786

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The Iwasawa invariants and the higher K-groups associated to real quadratic fields2005

    • Author(s)
      Sumida, Hiroki
    • Journal Title

      Experimental Mathematics vol.14

      Pages: 307-316

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On a theorem of Kawamuto on normal bases of rings of integers II2005

    • Author(s)
      市村 文男
    • Journal Title

      Canadian Mathematical Bulletin 48

      Pages: 576-579

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On the ring of p-integers of acyclic extension over a number field2005

    • Author(s)
      市村 文男
    • Journal Title

      Journal de Theorie des Numbres de Bordeaux 17

      Pages: 779-786

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On a theorem of Childs on normal bases of rings of integers : Addendum2004

    • Author(s)
      市村 文男
    • Journal Title

      Journal of London Mathematical Society 69

      Pages: 303-305

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Normal integral bases and ray class groups2004

    • Author(s)
      市村 文男
    • Journal Title

      Acta Arithmetica 114

      Pages: 71-85

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] On normal integral bases of unramified abelian p-extensions over a global function fields of characteristic p2004

    • Author(s)
      市村 文男
    • Journal Title

      Finite Fields and Their Apllications 10

      Pages: 432-437

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On a theorem of Kawamoto on normal bases of rings of integers2004

    • Author(s)
      市村 文男
    • Journal Title

      Tokyo Journal of Mathematics 27

      Pages: 527-540

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On a theorem of Childs on normal bases of rings of integers : Addendum2004

    • Author(s)
      Ichimura, Humid
    • Journal Title

      Journal of the London Mathematical Society vol.69

      Pages: 303-305

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Normal integral bases and ray class groups2004

    • Author(s)
      Ichimura, Humid
    • Journal Title

      Acta Arithmetica vol.114

      Pages: 71-85

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On normal integral bases of unramified abelian p-extensions over a global function field of characteristic p2004

    • Author(s)
      Ichimura, Humid
    • Journal Title

      Finite Fields and their Applications vol.10

      Pages: 432-437

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On a theorem of Kawamoto on normal bases of rings of integers2004

    • Author(s)
      Ichimura, Humid
    • Journal Title

      Tokyo Journal of Mathematics vol.27

      Pages: 527-540

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the ring of integers of a tame Kummer extension over a number field2004

    • Author(s)
      市村 文男
    • Journal Title

      Journal of Pure and Applied Algebra 187

      Pages: 169-182

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Addendum to "On a theorem of Childs on normal bases of rings of integers2004

    • Author(s)
      市村 文男
    • Journal Title

      Journal of the London Mathematical Society 69

      Pages: 303-305

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On normal integral bases of unramified abelian p-extension over a global function field of characteristic p2004

    • Author(s)
      市村 文男
    • Journal Title

      Finite Fields and their Applications 10

      Pages: 432-437

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On a theorem of Kawamoto on normal bases of rings of integers2004

    • Author(s)
      市村 文男
    • Journal Title

      Tokyo Journal of mathematics 27

      Pages: 527-540

    • Related Report
      2004 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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