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Applications of arithmetic theory of algebraic groups to computational number theory

Research Project

Project/Area Number 16540014
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionThe University of Electro-Communications

Principal Investigator

KIDA Masanari  The University of Electro-Communications, Faculty of Electro-Communications, Associate Professor, 電気通信学部, 助教授 (20272057)

Co-Investigator(Kenkyū-buntansha) OTA Kazuo  The University of Electro-Communications, Faculty of Electro-Communications, Professor, 電気通信学部, 教授 (80333491)
ONO Masahiro  The University of Electro-Communications, Faculty of Electro-Communications, Associate Professor, 電気通信学部, 助教授 (70277820)
TAYA Hisao  Tohoku University, Graduate School of Information Sciences, Assistant, 情報科学研究科, 助手 (40257241)
Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2004: ¥1,000,000 (Direct Cost: ¥1,000,000)
KeywordsKummer theory / algebraic number theory / computational number theory / トーラス / 素数判定法
Research Abstract

In this research we investigate a generalization of Kummer theory to fields without roots of unity. Kummer theory is a basic tool in algebra and number theory and has many applications in these areas. Our generalization uses commutative algebraic groups called norm tori. Under certain natural conditions, we prove a Kummer duality induced from a self-isogeny of norm tori. This is a natural extension of the classical Kummer theory. It also describes the cyclic extensions over certain fields without roots of unity.
As an application, we develop a method to compute cyclic polynomials arising from our Kummer theory and calculate some example of such polynomials using computer algebra system MAGMA. In the case of quintic cyclic polynomial, we can show a relationship between our Kummer polynomials and classical Lehmer polynomials. This enables us to give a complete description of the decomposition law in the cyclic quintic extensions.
As our result is quite general in the nature, we can expect more applications in the area of algebra and number theory.

Report

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

    (15 results)

All 2006 2005 2004 Other

All Journal Article (15 results)

  • [Journal Article] Cyclic polynomials arising from Kummer theory of norm algebraic tori2006

    • Author(s)
      Masanari Kida
    • Journal Title

      Lecture notes in Computer Science 4076

      Pages: 102-113

    • 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] Cyclic polynomials arising from Kummer theory of norm algebraic tori.2006

    • Author(s)
      Kida, Masanari
    • Journal Title

      Lecture Notes in Computer Science 4076

      Pages: 102-113

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Kummer theory for norm algebraic tori.2005

    • Author(s)
      Masanari Kida
    • Journal Title

      Journal of Algebra 293

      Pages: 427-447

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] 群と素数判定法2005

    • Author(s)
      木田 雅成
    • Journal Title

      仙台数論および組み合わせ論小研究集会2004報告集

      Pages: 77-86

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] 最小ベクトルの係数の存在範囲に関する考察2005

    • Author(s)
      金山直樹, 木田雅成, 太田和夫 他
    • Journal Title

      Proceedings of SCIS 2005

      Pages: 967-971

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] ノルム・トーラスのクンマー理論2005

    • Author(s)
      木田雅成
    • Journal Title

      京都大学数理解析研究所講究録 1451

      Pages: 237-242

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Kummer theory for norm algebraic tori.2005

    • Author(s)
      Kida, Masanari
    • Journal Title

      J.Algebra 293

      Pages: 427-447

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Kummer theory for norm algebraic tori2005

    • Author(s)
      Masanari Kida
    • Journal Title

      Journal of Algebra 293

      Pages: 427-447

    • Related Report
      2005 Annual Research Report
  • [Journal Article] ノルム・トーラスのクンマー理論2005

    • Author(s)
      木田 雅成
    • Journal Title

      京都大学数理解析研究所講究録 1451

      Pages: 237-242

    • Related Report
      2005 Annual Research Report
  • [Journal Article] 群と素数判定法2005

    • Author(s)
      木田 雅成
    • Journal Title

      仙台数論および組み合わせ論小研究集会報告集 (印刷中)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] 素数判定法について2005

    • Author(s)
      木田 雅成
    • Journal Title

      数理科学 43

      Pages: 19-24

    • Related Report
      2004 Annual Research Report
  • [Journal Article] 最小ベクトルの係数の存在範囲に関する考察2005

    • Author(s)
      金山直樹, 木田雅成, 太田和夫, 他
    • Journal Title

      Proceedings of SCIS 2005

      Pages: 967-971

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Primality tests using algebraic group2004

    • Author(s)
      Masanari Kida
    • Journal Title

      Experimental Math. 13

      Pages: 421-427

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Primality tests using algebraic group2004

    • Author(s)
      Masanari Kida
    • Journal Title

      Experimental Mathematics 13・4(印刷中)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] 冪根を含まない体のクンマー理論について

    • Author(s)
      木田雅成
    • Journal Title

      第5回北陸数論小研究集会報告集 (印刷中)

    • Related Report
      2006 Annual Research Report

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

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