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Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory
Lindsay N. Childs, State University of New York at Albany, NY
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Mathematical Surveys and Monographs
2000; 215 pp; hardcover
Volume: 80
ISBN-10: 0-8218-2131-8
ISBN-13: 978-0-8218-2131-2
List Price: US$63
Member Price: US$50.40
Order Code: SURV/80
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This book studies Hopf algebras over valuation rings of local fields and their application to the theory of wildly ramified extensions of local fields. The results, not previously published in book form, show that Hopf algebras play a natural role in local Galois module theory.

Included in this work are expositions of short exact sequences of Hopf algebras; Hopf Galois structures on separable field extensions; a generalization of Noether's theorem on the Galois module structure of tamely ramified extensions of local fields to wild extensions acted on by Hopf algebras; connections between tameness and being Galois for algebras acted on by a Hopf algebra; constructions by Larson and Greither of Hopf orders over valuation rings; ramification criteria of Byott and Greither for the associated order of the valuation ring of an extension of local fields to be Hopf order; the Galois module structure of wildly ramified cyclic extensions of local fields of degree \(p\) and \(p^2\); and Kummer theory of formal groups.

Beyond a general background in graduate-level algebra, some chapters assume an acquaintance with some algebraic number theory. From there, this exposition serves as an excellent resource and motivation for further work in the field.

Readership

Graduate students and research mathematicians interested in algebraic number theory, Hopf algebras and related topics.

Reviews

"This monograph, authored by a leading expert and main contributor in the field, is very painstakingly written ... The book offers an ideal road of entry into the subject, for everybody with an interest in local Galois module theory or in the theory of Hopf algebras."

-- Zentralblatt MATH

"Very well organized ... a pleasure to read ... excellent material for seminars or study groups ... offers an ideal road of entry into the subject."

-- Mathematical Reviews

Table of Contents

  • Introduction
  • Hopf algebras and Galois extensions
  • Hopf Galois structures on separable field extensions
  • Tame extensions and Noether's theorem
  • Hopf algebras of rank \(p\)
  • Larson orders
  • Cyclic extensions of degree \(p\)
  • Non-maximal orders
  • Ramification restrictions
  • Hopf algebras of rank \(p^2\)
  • Cyclic Hopf Galois extensions of degree \(p^2\)
  • Formal groups
  • Principal homogeneous spaces and formal groups
  • Bibliography
  • Index
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