Arithmetic of constructive inverse Galois problem

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540011
• 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 (2013-2014); The University of Electro-Communications (2012)
• Principal Investigator: KIDA Masanari 東京理科大学, 理学部, 教授 (20272057)
• Co-Investigator(Kenkyū-buntansha): OHNO Masahiro 電気通信大学, 情報理工学研究科, 准教授 (70277820)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: ガロア理論 / 生成多項式 / 代数的トーラス

Outline of Final Research Achievements

In this research we aimed at constructing families of polynomials with given finite groups as Glois group.
In particular, we were interested in families with arithmetic information such as ramification and ideal decomposition law. Our first object was Kummer extensions associated to algebraic tori. We made extensive computation on low dimensional tori case to find arithmetic information. I believe that it will soon become possible to get a generalization for certain tori of arbitrary dimensions. We obtained a new family of cyclic polynomials by using algebraic tori of norm tori. I submitted an article concerning this construction. I also made some investigation concerning dihedral extensions in the relation of complex multiplication.

Free Research Field

整数論