Abhyankar's lemma and the class group
Abhyankar's lemma and the class group
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Abhyankar 引理和类群
DOI：
10.1007/bfb0062703
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发表时间：
1979
期刊：
影响因子：
--
通讯作者：
G. Cornell
中科院分区：
文献类型：
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作者：
G. Cornell
我们通过类场论知道。数域E的分歧阿贝尔扩张对应于E的类群的商群，因此通过对偶（非正则）对应于E的类群的子群。因此，要找到数域E的例子，其类群具有某些性质（大秩，大阶元素等）。找到具有非分歧阿贝尔扩张的数域E，其伽罗瓦群具有所需的性质就足够了。虽然这通常是困难的，但有一种方法可以做到这一点：假设E包含一个子域F，其中F和它的阿贝尔扩张在某种程度上是”更好地了解”的，那么我们可以问是否存在F的阿贝尔扩张，比如F* v1与E复合，EF是E的非分歧阿贝尔扩张？当F* 被取为极大的这样的域时，我们称E相对于ε的亏格域。如果F是Q，我们就说绝对亏格场。这些领域的研究是由Frohlich在两个开创性的论文[Fro 1，Fro 2]，后来研究了古田（毛皮1]和石田[Ishl]除其他外。本说明使用Abhyankar引理表明，某些类别的领域有有趣的属领域相对于一个适当选择的子域。例如，我们证明了任何阿贝尔群都是分圆域的类群的子群。子域的选择是重要的，因为可以证明，相对于子域的“wr· ong”选择，域E可能没有适当更大的亏格域。例如，有一个经典的结果，由于利奥波德说，该领域的第n次根的单位，Q（z; n）没有一个适当的更大的绝对亏格领域。对于所有这一切（和更多）见石田的专着（伊什1）。由于证明Abhyankar的引理是很难找到我已经采取了自由，包括一个在这里。这取决于
We know by class field theory that ur. ramified abelian extensions of a number field E correspond to quotient groups of the class group of E and so via duality (non-canonically) to subgroups of the class group of E. Thus to find examples of number fields E whose class groups have certain properties (large rank, elements of large order, etc.) it would be enough to find number fields E with unramified abelian extensions whose galois groups have the required properties. While this is usually difficult, one way to do this is the following: Suppose E contains a subfield F where F and its abelian extensions are somehow" better known" we can then ask if there is an abelian extension of F, say F* v1hose composites withE, EF is an unramified abelian extension of E? When F* is taken to be the maximal such field, we speak of the Genus Field of E relative to£. If F is Q we speak of the absolute Genus Field. The study of these fields was initiated by Frohlich in two seminal papers [Fro l, Fro 2] and later studied by Furuta (Fur 1] and Ishida [Ish l] among others.The present note uses Abhyankar's lemma to show that certain classes of fields have interesting genus fields relative to a properly chosen subfield. For example we show that any abelian group is a subgrcup of the class group of a cyclotomic field. The choice of subfield is important, for it is possible to show that a field E may not have a properly larger genus field relative to the'wr· ong'choice of subfield. For example there is a classical result due to Leopoldt that says that the field of n'th roots of unity, Q (z; n) hasn't a properly larger absolute genus field. For all this (and more) see Ishida's monograph (Ish 1]. Since proofs of Abhyankar's lemma are hard to find I have taken the liberty of including one here. It depends on a