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E-raamat: Hopf Algebras and Galois Module Theory

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Hopf Algebras and Galois Module TheorySuurem pilt
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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.
Chapter 1 Introduction: What is this book about?
1(10)
1.1 Background: Hopf-Galois theory and Galois module theory
1(3)
1.2 Hopf-Galois structures since 2000
4(2)
1.3 Galois module theory since 2000
6(3)
1.4 What's not in this book
9(1)
Acknowledgments
10(1)
Part I Hopf-Galois Extensions
11(152)
Chapter 2 Hopf-Galois structures on Galois extensions of fields, regular subgroups, and skew braces
13(14)
2.1 Introduction
13(1)
2.2 Greither-Pareigis theory
14(2)
2.3 Byott translation theory
16(2)
2.4 Actions by the left regular representations
18(1)
2.5 Counting
19(1)
2.6 Working with regular subgroups of Hol(N)
19(3)
2.7 Radical algebras
22(1)
2.8 Skew (left) braces
23(1)
2.9 Connecting skew braces with Hopf-Galois structures
24(1)
2.10 Isomorphic skew braces
25(2)
Chapter 3 (Non)-existence results on Hopf-Galois structures
27(14)
3.1 Introduction
27(1)
3.2 p-groups, p an odd prime, cyclic case
28(2)
3.3 2-groups
30(1)
3.4 Groups of composite order n that decompose
31(2)
3.5 Cases where G must be isomorphic to N
33(2)
3.6 Realizability when G = Sn or An
35(1)
3.7 Cases where given G, N can be any group with |G| = |N|
35(1)
3.8 If G is abelian or nilpotent, then TV is?
36(1)
3.9 Kohl's non-existence theorem
37(1)
3.10 The case G metabelian and radical algebras
38(1)
3.11 Other realizability results
39(2)
Chapter 4 Hopf-Galois structures arising from fixed point free pairs of homomorphisms
41(18)
4.1 Introduction
41(1)
4.2 Fixed point free pairs of homomorphisms
42(1)
4.3 The case N = G
43(3)
4.4 The action of L[ N]G on L
46(2)
4.5 Fixed point free endomorphisms
48(2)
4.6 Examples with N ≠ G
50(4)
4.7 Bi-skew braces and semidirect products
54(2)
4.8 Bi-skew braces and nilpotent rings
56(3)
Chapter 5 Quantitative results
59(10)
5.1 Introduction
59(1)
5.2 Regular subgroups and nilpotent algebras
60(1)
5.3 Elementary abelian p-groups
61(1)
5.4 Almost trivial algebras
62(3)
5.5 Asymptotic results on e(Cnp,Cnp) for large n
65(1)
5.6 Other counting results
66(3)
Chapter 6 Enumeration of Hopf-Galois structures on Galois extensions of degree mp
69(14)
6.1 Introduction
69(1)
6.2 Preliminaries
70(1)
6.3 Twisted wreath products
71(2)
6.4 Enumeration within Smp
73(5)
6.5 Block systems and Hopf-Galois structures
78(5)
Chapter 7 On the Galois correspondence for Hopf-Galois structures
83(20)
7.1 Introduction
83(2)
7.2 k-Hopf subalgebras
85(2)
7.3 On the Galois correspondence for Hopf-Galois structures
87(2)
7.4 Kohl's application of Corollary 7.6
89(1)
7.5 A skew brace setting
90(2)
7.6 Fixed point free pairs
92(3)
7.7 Radical algebras and the Galois correspondence
95(2)
7.8 Commutative examples
97(1)
7.9 Elementary abelian p-groups
98(1)
7.10 Non-normal Hopf-Galois structures
99(4)
Chapter 8 Normality in Hopf-Galois extensions
103(14)
8.1 Normality for Hopf-Galois structures on Galois extensions
104(4)
8.2 Skew braces and normality in Hopf-Galois extensions
108(3)
8.3 Induced Hopf-Galois structures
111(6)
Chapter 9 Descent theory, and the structure of Hopf algebras acting on separable field extensions
117(28)
9.1 General descent theory
118(8)
9.2 Galois descent for Hopf algebras
126(16)
9.3 Absolutely semisimple forms
142(3)
Chapter 10 Hopf-Galois actions on purely inseparable extensions
145(18)
10.1 A little bit of algebraic geometry
145(7)
10.2 A short history
152(7)
10.3 Hopf-Galois structures on modular extensions
159(4)
Part II Hopf-Galois Module Theory
163(134)
Chapter 11 Hopf-Galois module theory
165(24)
11.1 The Normal Basis Theorem for Hopf-Galois structures
166(3)
11.2 Hopf orders and Childs' theorem
169(5)
11.3 Associated orders for opposite Hopf-Galois structures
174(1)
11.4 Subextension techniques
175(4)
11.5 Tamely ramified extensions
179(4)
11.6 Extensions of number fields
183(6)
Chapter 12 Hopf orders in group rings
189(48)
12.1 Hopf orders and Galois module theory
189(2)
12.2 Dual Hopf orders
191(4)
12.3 Byott's theorem on realizability
195(2)
12.4 Group valuations and Larson orders
197(7)
12.5 Hopf orders in K[ G], G = Cp
204(6)
12.6 Hopf orders in K[ G], G = Cp × Cp, G = Cp2
210(3)
12.7 Hopf orders in K[ G], G = CpS
213(2)
12.8 General constructions in the cases G -- Cpn, G = Cp
215(1)
12.9 Truncated exponential Hopf orders
216(2)
12.10 Models of μpn for n = 1,2,3
218(1)
12.11 Hopf orders and realizability
219(5)
12.12 Realizable Hopf orders in K[ Cpn], K[ Cnp]
224(1)
12.13 When K has characteristic p
225(12)
Chapter 13 Ramification theory for separable extensions of local fields
237(18)
13.1 Basic theory
237(3)
13.2 Power series and Herbrand's theorem
240(7)
13.3 Properties of lower ramification breaks
247(8)
Chapter 14 Stable and semistable Hopf-Galois extensions
255(30)
14.1 Bondarko's map ip for Hopf-Galois extensions
255(6)
14.2 Some technical lemmas
261(5)
14.3 Diagrams of elements of L × k L
266(5)
14.4 H-stable and H-semistable extensions
271(2)
14.5 Hopf-Galois module structure
273(7)
14.6 A non-classical example
280(5)
Chapter 15 Hopf-Galois scaffolds
285(12)
15.1 H-scaffolds
285(1)
15.2 Examples of H-scaffolds
286(3)
15.3 Some basic properties of H-scaffolds
289(2)
15.4 H-semistable extensions and H-scaffolds
291(2)
15.5 Ramified extensions of degree p
293(4)
Bibliography 297(12)
Index 309
Lindsay N. Childs, University at Albany, NY.

Cornelius Greither, Universitat der Bundeswehr Munchen, Neubiberg, Germany.

Kevin P. Keating, University of Florida, Gainesville, FL.

Alan Koch, Agnes Scott College, Decatur, GA.

Timothy Kohl, Boston University, MA.

Paul J. Truman, Keele University, Staffordshire, United Kingdom.

Robert G. Underwood, Auburn University at Montgomery, AL.
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