On Fields of Class Two
On Fields of Class Two
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论二类领域
DOI：
10.1112/plms/s3-4.1.235
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发表时间：
1954
影响因子：
1.8
通讯作者：
A. Fröhlich
中科院分区：
文献类型：
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作者：
A. Fröhlich
具有终止中心级数的有限群的类，即有限幂零群的类，是该级数的长度（参见(7)）。因此，中心包含其对易子群的非阿贝尔群属于第二类。用幂零伽罗瓦群的类定义了一类具有幂零伽罗瓦群的自共轭代数数域。本文主要讨论有理域上最多二类的域。它的主要目的是确定所有这些领域的集合以及这一集合中的基本关系和操作。判定标准本质上是理性的，可以用理性同余关系来表示。这种关系也将被用来确定伽罗瓦群，以及每个这样的场的惯性群。用类场论解决了至多为一类场的相应问题集。因此，本文给出了一类有理域的所有可能的非阿贝尔扩展的一个理论。在其他论文中，我将把这些结果应用于阿贝尔域的绝对类群的研究和素分解的非阿贝尔定律的研究。最多二类的群是最多二类的素数幂群的直接积。因此，只考虑其度数是任意但固定的素数I的幂的域就足够了。为了不使论证超载，我将明确地假设我是奇数，只要这是出于简洁的原因所必需的。虽然我不会在每个地方都给出I= 2的证明，但我将指出特殊情况下论证的主要变化，并且我将不受I的限制地陈述定理。二次互易律将作为一个一般定理的必然结果得到；这可以看作是对在I= 2的情况下所遇到的特殊困难的补偿。
THE class of a finite group with terminating central series, ie of a finite nilpotent group, is the length of that series (cf.(7)). Thus a non-Abelian group whose central contains its commutator group is of class two. I define the class of a self-conjugate algebraic number field with nilpotent Galois group by the class of that group. In the present paper we are concerned with fields of at most class two over the rational field. Its principal aim is the determination of the set of all such fields and the fundamental relations and operations in this set. The criteria of determination are essentially rational, expressible in terms of rational congruence relations. Such relations will also be shown to determine the Galois group, and the inertia groups of each such field. The corresponding set of problems for fields of at most class one has been solved by class field theory. The present paper thus gives a theory of all possible non-Abelian extensions of the rational field which are of olass two. In other papers I shall apply these results to a study of the absolute class group of Abelian fields and to an investigation of non-Abelian laws of prime decomposition.A group of at most class two is the direct product of prime power groups of at most class two. It is thus sufficient to consider only fields whose degree is the power of an arbitrary, but fixed, prime I. The case I= 2 requires repeatedly special discussion. In order not to overload the argument I shall explicitly assume I to be odd whenever that seems necessary for reasons of conciseness. Though I shall not everywhere give the proofs for I= 2, 1 shall indicate the principal alterations in argument for the special case, and I shall state the theorems without restriction on I. The quadratic law of reciprocity will be obtained as the corollary of one of the general theorems; this may be considered as a compensation for the particular difficulties encountered in the case of I= 2.