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.
Rezensionen / Stimmen
"...a clearly written book, which uses (almost) exclusively algebraic language (and no cohomology), and which will be useful for every algebraist or number theorist. It is easily accessible and suitable also for first-year graduate students." Mathematical Reviews
Reihe
Sprache
Verlagsort
Zielgruppe
Illustrationen
Worked examples or Exercises; 1 Tables, unspecified; 7 Line drawings, unspecified
Maße
Höhe: 244 mm
Breite: 160 mm
Dicke: 20 mm
Gewicht
ISBN-13
978-0-521-81998-5 (9780521819985)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Klassifikation
Autor*in
University of Copenhagen
Texas Tech University
Queen's University, Ontario
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.
