2004 Fiscal Year Final Research Report Summary
The properties of the group of units and the Leopoldt conjecture
| Project/Area Number |
13640052
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Oyama National College of Technology |
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Principal Investigator |
SHIMADA Tsutomu Oyama National College of Technology, Department of General Education, Professor, 一般科, 教授 (40321393)
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| Project Period (FY) |
2001 – 2004
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| Keywords | The Leopoldt conjecture / The Group of Units / The Fermat Quotient |
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| 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.
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