2004 Fiscal Year Final Research Report Summary

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 Type	Single-year Grants
Section	一般
Research Field	Algebra
Research Institution	Oyama 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
Keywords	The Leopoldt conjecture / The Group of Units / The Fermat Quotient
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.

(4 results)