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The properties of the group of units and the Leopoldt conjecture

单位群的性质和利奥波德猜想

基本信息

批准号：
13640052
负责人：
SHIMADA Tsutomu
金额：
$ 0.64万
依托单位：
Oyama National College of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2004
项目状态：
已结题

项目摘要

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.

我们研究了Leopoldt猜想和一些相关问题。在代数上证明了Brumer的一个定理，得到了以下两个结果。首先，设k是有限次代数数域，K是k的有限阿贝尔扩张，p(&gt；3)是奇素数。如果p的Leopoldt猜想对K/k的所有循环子扩张都成立，那么这个猜想对K也成立，这是Miki首先利用Galois群的结构证明的。我们的证明是从不使用伽罗瓦群的结构的角度来研究的。其次，我们证明了数域k在有理数域上的3次Galois扩张的猜想。我们的结果推广了Miki证明的定理q=3：当p和q是奇素数，k是有理数域上q次的循环扩张，p是模q的原根，则Leopoldt猜想对k和p成立。