Swan Modules and Hilbert–Speiser Number Fields
Swan Modules and Hilbert–Speiser Number Fields
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Swan 模和 Hilbert-Speiser 数域
DOI：
10.1006/jnth.1999.2425
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发表时间：
1999
影响因子：
0.7
通讯作者：
A. Srivastav
中科院分区：
文献类型：
--
作者：
C. Greither;Daniel R. Replogle;K. Rubin;A. Srivastav
摘要 如果对于每个驯化的有限阿贝尔扩张 N/K，N、O N 的代数整数环在 K 的代数整数环 O K 上具有正规积分基，则数域 K 被称为希尔伯特-斯佩塞域。经典的希尔伯特-斯佩塞定理证明有理数域 Q 是这样一个域。众所周知，Hilbert-Speiser 域的类数必须等于 1。我们考虑数域 K 的基本阿贝尔扩展和 Swan 模，以获得 K 是 Hilbert-Speiser 域的附加必要条件。我们证明，在所有数域中，Q 域是唯一的 Hilbert-Speiser 域。
Abstract A number field K is called a Hilbert–Speiser field if for each tamely ramified finite abelian extension N/K the ring of algebraic integers of N, O N, has a normal integral basis over O K, the ring of algebraic integers of K. The classical Hilbert–Speiser theorem proves that the field of rational numbers Q is such a field. It is well known that the class number of a Hilbert–Speiser field must equal 1. We consider tame elementary abelian extensions of a number field K and Swan modules to obtain additional necessary conditions for K to be a Hilbert–Speiser field. We show that among all number fields, the field Q is the only Hilbert–Speiser field.