一般多項式：逆ガロア問題の構成<br>Generic Polynomials : Constructive Aspects of the Inverse Galois Problem (Mathematical Sciences Research Institute Publications, 45)

一般多項式：逆ガロア問題の構成
Generic Polynomials : Constructive Aspects of the Inverse Galois Problem (Mathematical Sciences Research Institute Publications, 45)

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  • 製本 Hardcover:ハードカバー版／ページ数 258 p.
  • 言語 ENG
  • 商品コード 9780521819985
  • DDC分類 512.3

Full Description


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.

Contents

Introduction; 1. Preliminaries; 2. Groups of small degree; 3. Hilbertian fields; 4. Galois theory of commutative rings; 5. Generic extensions and generic polynomials; 6. Solvable groups I: p-groups; 7. Solvable groups II: Frobenius groups; 8. The number of parameters; Appendix A. Technical results; Appendix B. Invariant theory; Bibliography; Index.