Generic Polynomials
Constructive Aspects of the Inverse Galois Problem
Seiten
2002
Cambridge University Press (Verlag)
978-0-521-81998-5 (ISBN)
Cambridge University Press (Verlag)
978-0-521-81998-5 (ISBN)
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The main theme in this book is a constructive approach to the Inverse Galois problem. The existence of certain 'Generic' polynomials, which give all Galois extensions having the required group as their Galois group, is demonstrated and where they do exist, a detailed treatment of their construction is given.
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.
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.
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.
| Erscheint lt. Verlag | 9.12.2002 |
|---|---|
| Reihe/Serie | Mathematical Sciences Research Institute Publications |
| Zusatzinfo | Worked examples or Exercises; 1 Tables, unspecified; 7 Line drawings, unspecified |
| Verlagsort | Cambridge |
| Sprache | englisch |
| Maße | 160 x 244 mm |
| Gewicht | 509 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie | |
| ISBN-10 | 0-521-81998-9 / 0521819989 |
| ISBN-13 | 978-0-521-81998-5 / 9780521819985 |
| Zustand | Neuware |
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