A note on class numbers of algebraic number fields
A note on class numbers of algebraic number fields
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关于代数数域的类数的注记
DOI：
10.1007/bf03374563
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发表时间：
1956
期刊：
影响因子：
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通讯作者：
K. Iwasawa
中科院分区：
文献类型：
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作者：
K. Iwasawa
I的证明如下：设A和A'分别是k和k的第m个最大的无分支阿贝尔扩展。由类场论[A: k]= hand [A': Kl= H.由于P被k /k完全分形，我们得到k f\A= k， KAlk的伽罗瓦群是KAlk的伽罗瓦群与KAIA的伽罗瓦群的直接积。那么很容易得出KA是K的非分支阿贝尔扩展，因此它包含在A'中。
The proof of I is as follows2): Let A and A'be the m: 1ximal unramified abelian extensions of k and K respectively. By the class field theory,[A: k]= hand [A': Kl= H. Since P is fully ramificd by K/k, we have K f\A= k and the Galois group of KAlk is the direct product of the Galois groups of KAlK and of KAIA. It then follows easily that KA is an unramified abelian extension of K and, hence, is contained in A'.