On the Stickelberger ideal and the circular units of a cyclotomic field
On the Stickelberger ideal and the circular units of a cyclotomic field
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关于斯蒂克伯格理想和分圆场的圆形单位
DOI：
10.2307/1970932
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发表时间：
1978
影响因子：
4.9
通讯作者：
W. Sinnott
中科院分区：
文献类型：
--
作者：
W. Sinnott
所谓分环场，是指复数C在有理数Q上由单位根生成的子场。设k是一个假想的分环场。令Cn = e2ri/"对于任意整数n > 1。那么存在一个唯一的整数m >2, m t2 mod 4，使得k Q(Qm)；我们称m为k的导体。本文考虑与k有关的两个对象：stickkelberger理想S和圆单位c。设G为k / Q的伽罗瓦群，设R = Z[G]为G在普通整数Z上的群环。C是单位群e (k)的一个子群。设j表示由复共轭引起的G的元素。如果A是任意g模，我们用A+表示A中ja= A的元素集合A，用A表示A中ja= -a的元素集合A。我们的主要结果将是索引[R-: S-]和[E+: C+]的计算。现在我们精确地陈述我们的结果。设h表示k的类数，h+ k+的类数（k的最大全实数子域），设hh/h+。然后我们证明如下。
By a cyclotomic field, we shall mean a subfield of the complex numbers C generated over the rational numbers Q by a root of unity. Let k be an imaginary cyclotomic field. Let Cn = e2ri/" for any integer n > 1. There is then a unique integer m > 2, m t 2 mod 4, such that k Q(Qm); we call m the conductor of k. We consider in this paper two objects associated with k: the Stickelberger ideal S and the circular units C. Let G be the Galois group of k over Q, and let R = Z[G] be a group ring of G over the ordinary integers Z. S is then an ideal of R; C is a subgroup of the group of units Eof k. Let j denote the element of G induced by complex conjugation. If A is any G-module, we denote by A+ the set of elements a in A for which ja = a, and by Athe set of elements a in A for which ja= -a. Our main result will be a computation of the indices [R-: S-] and [E+: C+]. We now state our result precisely. Let h denote the class number of k, h+ the class number of k+ (the maximal totally real subfield of k), and put hh/h+. Then we prove the following.