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The properties of the group of units and the Leopoldt conjecture

Research Project

Project/Area Number 13640052
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionOyama National College of Technology

Principal Investigator

SHIMADA Tsutomu  Oyama National College of Technology, Department of General Education, Professor, 一般科, 教授 (40321393)

Project Period (FY) 2001 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2004: ¥200,000 (Direct Cost: ¥200,000)
Fiscal Year 2003: ¥200,000 (Direct Cost: ¥200,000)
Fiscal Year 2002: ¥200,000 (Direct Cost: ¥200,000)
Fiscal Year 2001: ¥400,000 (Direct Cost: ¥400,000)
KeywordsThe Leopoldt conjecture / The Group of Units / The Fermat Quotient / 単数群 / 円分体 / 類数 / 二次体 / 虚二次体
Research Abstract

We investigated the Leopoldt conjecture and some related topics. On the algebraic another proof of a theorem of Brumer, we get following two results. First, let k be an algebraic number field of finite degree, K a finite abelian extension of k and p(>3) an odd prime number. If the Leopoldt conjecture for p is valid for all cyclic subextensions of K/k, then the conjecture is also true for K. This is first proved by Miki by using the structure of the galois group. Our proof is studied from the point of view of no use of the structure of the galois groups. Next, we proved the conjecture for the number field k the galois extension of degree 3 over the rational number field. Our result is a generalization for q=3 of the theorem proved by Miki : when p and q be odd primes, k be a cyclic extension of degree q over the rational number field and p is a primitive root modulo q, then the Leopoldt conjecture holds for k and p.

Report

(5 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (7 results)

All 2004 Other

All Journal Article (6 results) Publications (1 results)

  • [Journal Article] Another Proof of a Lemma on the Leopoldt Conjecture2004

    • Author(s)
      T.Shimada
    • Journal Title

      Tokyo Metropolitan University Mathematics Preprint Series No.9

      Pages: 1-3

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] レオポルド予想について2004

    • Author(s)
      T.Shimada
    • Journal Title

      第2回北陸数論研究集会報告集 1

      Pages: 35-48

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Another Proof of a Lemma on the Leopoldt Conjecture2004

    • Author(s)
      Tsutomu Shimada
    • Journal Title

      Tokyo Metropolitan University Mathematics Preprint Series No.9

      Pages: 1-3

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the Leopoldt conjecture2004

    • Author(s)
      Tsutomu Shimada
    • Journal Title

      Proceedings of the 2^<nd> Workshop on Number Theory, Kanazawa University

      Pages: 35-48

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Another Proof of a Lemma on the Leopoldt Conjecture2004

    • Author(s)
      Tsutomu Shimada
    • Journal Title

      Tokyo Metropolitan University Mathematics Preprint Series 2004:No.9

      Pages: 1-3

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Leopoldt予想について2004

    • Author(s)
      島田 勉
    • Journal Title

      第2回北陸数論研究集会報告集 1

      Pages: 35-48

    • Related Report
      2004 Annual Research Report
  • [Publications] Tsutomu Shimada: "Another Proof of a Lemma on the Leopoldt Conjecture"Tokyo Metropolitan University Mathematics Preprint Series. 2004 No.9. 1-3 (2004)

    • Related Report
      2003 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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