Research on the reciprocity law of local fields

Research Project

Project/Area Number	09640045
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Single-year Grants
Section	一般
Research Field	Algebra
Research Institution	Kagawa University
Principal Investigator	NAITO Hirotada Kagawa University of Education, A.P., 教育学部, 助教授 (00180224)
Co-Investigator(Kenkyū-buntansha)	NAKAJIMA Shoichi Gakushuin, Science, P., 理学部, 教授 (90172311)
Project Period (FY)	1997 – 1999
Project Status	Completed (Fiscal Year 1999)
Budget Amount *help	¥2,500,000 (Direct Cost: ¥2,500,000) Fiscal Year 1999: ¥800,000 (Direct Cost: ¥800,000) Fiscal Year 1998: ¥700,000 (Direct Cost: ¥700,000) Fiscal Year 1997: ¥1,000,000 (Direct Cost: ¥1,000,000)
Keywords	Number Theo / local field
Research Abstract	We determine all extensions over the rational p-adic fields which are generated by 3 -division points of elliptic curves, We see that there are three fields which are not generated by such points. As an application, we construct algebraic number fields whose Galois group over the rational number field are isomorphic to the general linear group of degree two over the prime field of three elements with some local conditions ( decomposition law). We construct infinitely many imaginary quadratic extensions over the fixed totally real number fields such that their relative class numbers are indivisible by the fixed odd prime with some loca conditions. We rediscover algebraic number fields such 'that their Dedekind zeta functions coincide but they are not conjugate, in the Galois extentions over the rational number fields whose Galois group over the rational number field are isomorphic to the general linear group of degree two over the prime field of p elements. Moreover we prove that the non p parts of the ideal class groups are isomorphic each other.