Computational study on non-abelian extensions

2019 Fiscal Year Final Research Report

• Project/Area Number: 15K04798
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo University of Science
• Principal Investigator: Kida Masanari 東京理科大学, 理学部第一部数学科, 教授 (20272057)
• Project Period (FY): 2015-10-21 – 2020-03-31
• Keywords: ガロア群 / ガロア拡大 / 同質類 / 非アーベル拡大 / 逆ガロア問題

Outline of Final Research Achievements

The first aim of the research was constructing non-abelian extensions with simple arithmetic
by using computational number theory, but in the course of research I found that the notion of isoclinism of finite groups helps a lot to study and construct extensions of number fields and the result is more fruitful than expected. We studied the smallest isoclinism class of non-abelian group and found that they several important common properties and also studied the isoclinism class containing the dihedral group of order 8 and found some significant difference between the classes. These results develop a new method of studying Galois extensions.

Free Research Field

整数論

Academic Significance and Societal Importance of the Research Achievements

ガロア群が非アーベル群になる非アーベル拡大の研究は整数論にとっての大きな課題である．保形表現との関連の研究が大いに進んでいるが，この研究では，ガロア群の同質類という同型類よりも弱い類別に着目し，同じ同質類にはいる拡大体の類似，異なる同質類にはいる拡大体の差異などを主に代数的な手法を使って研究した．このような手法を使った研究はこれまでになく，さらに幾つかの興味深い結果も得られていることから，今後もこの方法は有効に活用されていくものと考えている．