Class numbers and ℤp-extensions
Class numbers and ℤp-extensions
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类号和 ℤp-扩展名
DOI：
10.1007/bf01352651
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发表时间：
1975
影响因子：
1.4
通讯作者：
L. Washington
中科院分区：
文献类型：
--
作者：
L. Washington
设k是有理数域~上的有限次代数数域，K/k是一个Zp-扩张，一个Galois群同构于p-进整数的加群Zp(p是素数)的Galois扩张.对每个整数n>；0，都有一个唯一的子域k，它的p次K“在k上.设h是k的类数，PEN是p除以h的精确幂.岩泽已经证明了存在与n无关的整数2，/~和v，使得对于所有足够大的n，e，=2，+/p“+v。人们很自然地会问，是否可以说类似的事情可以关于h除以h的幂，其中f是不等于p的素数。本文的目的就是研究这个问题。
Let k be an algebraic number field of finite degree over the rational field~ and let K/k be a Zp-extension" a Galois extension with Galois group isomorphic to the additive group Zp of p-adic integers (p is a prime number). For each integer n> 0, there is a unique subfield k, of K of degree p" over k. Let h, be the class number of k, and let pen be the exact power of p dividing h,. Iwasawa has proved that there exist integers 2,/~, and v, independent of n, such that e,= 2,+/~ p"+ v for all sufficiently large n. It is natural to ask whether or not anything similar can be said about the power of¢ dividing h,, where f is a prime not equal to p. The purpose of this paper is to investigate this question.