Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory: Hopf Algebras and Local Galois Module Theory

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American Mathematical Soc., 2000 - Mathematics - 215 pages
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.
 

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Contents

Chapter 0 Introduction
1
Chapter 1 Hopf algebras and Galois extensions
7
Chapter 2 Hopf Galois structures on separable field extensions
47
Chapter 3 Tame extensions and Noethers Theorem
73
Chapter 4 Hopf algebras of rank p
85
Chapter 5 Larson orders
97
Chapter 6 Cyclic extensions of degree p
113
Chapter 7 Nonmaximal orders
129
Chapter 8 Ramification restrictions
135
Chapter 9 Hopf algebras of rank psup2
149
Chapter 10 Cyclic Hopf Galois extensions of degree psup2
161
Chapter 11 Formal groups
171
Chapter 12 Principal homogeneous spaces and formal groups
191
Bibliography
205
Index
213
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