The properties of the group of units and the Leopoldt conjecture

我们研究了利奥波德猜想和一些相关的话题。对Brumer定理的另一个代数证明，得到了如下两个结果。首先，设k是有限次的代数数域，k是k的有限阿贝尔扩展，p（>3）是奇素数。如果p的利奥波德猜想对K/ K的所有循环子扩展都成立，则该猜想对K也成立。这是Miki首先利用伽罗瓦群的结构证明的。我们的证明是从不使用伽罗瓦群结构的角度来研究的。其次，证明了数域k在有理数域上3次的伽罗瓦推广的猜想。我们的结果是对Miki证明的q=3定理的推广：当p和q是奇素数，k是有理数域上q次的循环扩展，p是原根模q时，则k和p成立利奥波德猜想。

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.