Hopf Algebras and Galois Module Theory

ISBN-13: 9781470465162

Lindsay N. Childs;Cornelius Greither;Kevin P. Keating

Opis

Hopf algebras have been shown to play a natural role in studying questions of integral module structure in extensions of local or global fields. This book surveys the state of the art in Hopf-Galois theory and Hopf-Galois module theory and can be viewed as a sequel to the first author's book, Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory, which was published in 2000. The book is divided into two parts. Part I is more algebraic and focuses on Hopf-Galois structures on Galois field extensions, as well as the connection between this topic and the theory of skew braces. Part II is more number theoretical and studies the application of Hopf algebras to questions of integral module structure in extensions of local or global fields. Graduate students and researchers with a general background in graduate-level algebra, algebraic number theory, and some familiarity with Hopf algebras will appreciate the overview of the current state of this exciting area and the suggestions for numerous avenues for further research and investigation.

Spis treści

• Introduction: What is this book about?
• Hopf-Galois extensions: Hopf-Galois structures on Galois extensions of fields, regular subgroups, and skew braces
• (Non)-existence results on Hopf-Galois structures
• Hopf-Galois structures arising from fixed point free pairs of homomorphisms
• Quantitative results
• Enumeration of Hopf-Galois structures on Galois extenion of degree $mp$
• On the Galois correspondence for Hopf-Galois structures
• Normality in Hopf-Galois extensions
• Descent theory, and the structure of Hopf algebras acting on separable field extensions
• Hopf-Galois actions on purely inseparable extensions
• Hopf-Galois module theory: Hopf-Galois module theory
• Hopf orders in group rings
• Ramification theory for separable extensions of local fields
• Stable and semistable Hopf-Galois extensions
• Hopf-Galois scaffolds
• Bibliography
• Index