Generic Polynomials: Constructive Aspects of the Inverse Galois Problem
Generic Polynomials: Constructive Aspects of the Inverse Galois Problem
复制标题
泛型多项式：伽罗瓦反问题的构造性方面
DOI：
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
由井 典子
中科院分区：
文献类型：
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作者：
C. U. Jensen;A. Ledet;由井 典子
这本书描述了一种构造伽罗瓦反问题的方法：给定一个有限群G和一个域K，确定是否存在其伽罗瓦群与G同构的K的伽罗瓦扩张。此外，如果存在这样的伽罗瓦扩张，则在K上找到其伽罗瓦群是指定群G的显式多项式。本书的主要主题是对某些有限群的�通用�多项式的族的论述，它给出了所有具有所需群作为其伽罗瓦群的伽罗瓦扩张。讨论了这种一般多项式的存在性，并在它们确实存在的情况下，详细地讨论了它们的构造。这本书还介绍了�通用维度�的概念，以解决通用多项式所需的最小参数数量的问题。
This book describes a constructive approach to the inverse Galois problem: Given a finite group G and a field K, determine whether there exists a Galois extension of K whose Galois group is isomorphic to G. Further, if there is such a Galois extension, find an explicit polynomial over K whose Galois group is the prescribed group G. The main theme of the book is an exposition of a family of �generic� polynomials for certain finite groups, which give all Galois extensions having the required group as their Galois group. The existence of such generic polynomials is discussed, and where they do exist, a detailed treatment of their construction is given. The book also introduces the notion of �generic dimension� to address the problem of the smallest number of parameters required by a generic polynomial.