Distribution of prime factors of the class numbers of prime-power cyclotomic fields

• Project/Area Number: 26400020
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Gakushuin University
• Principal Investigator: NAKAJIMA SHOICHI 学習院大学, 理学部, 教授 (90172311)
• Research Collaborator: Taniguchi Tetsuya
• Project Period (FY): 2014-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 円分体 / 類数 / 円分体の類数

Outline of Final Research Achievements

Class numbers of cyclotomic fields are important object of research in Algebraic Number Theory. As to the computation of them, cyclotomic fields of prime conductors were main target of research. In this project, we enlarged the area of computation to (minus parts of) class numbers of cyclotomic fields of prime power conductors. We focused on the Hypothesis, posed by Prof. H.Ichimura, that the relative minus-part class numbers of the prime power cyclotomic fields are relatively prime to each other for all prime powers.
On our project, we performed an extensive computation of the class numbers for prime-power cyclotomic fields, and verified that the above Hypothesis is (almost) valid in the range of our computation, finding only one exception to the Hypothesis.

Academic Significance and Societal Importance of the Research Achievements

フェルマーの最終定理への応用があることからも分かるように、円分体の類数は、代数的整数論において非常に重要な役割を果たす数である。しかし、従来の類数の計算においては、素数分体の場合が主眼であったし、類数の素因数について考察されることも少なかった。本研究では、素数ベキ分体の場合に研さん範囲を広げ、類数の素因子について新しい仮説の検証を行った。これは、類数の素因子、というテーマについて新しい局面を切り開くものである。