Taming wild extensions : Hopf algebras and local Galois module theory / Lindsay N. Childs.

Format:

Book

Author/Creator:

Childs, Lindsay, author.

Series:

Mathematical surveys and monographs ; no. 80.

Mathematical surveys and monographs, 0076-5376 ; volume 80

Language:

English

Subjects (All):

Hopf algebras.

Galois modules (Algebra).

Field extensions (Mathematics).

Physical Description:

1 online resource (225 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2000]

Language Note:

English

Summary:

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 p2; 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.

Contents:

""Contents""; ""Chapter 0. Introduction""; ""Chapter 1. Hopf algebras and Galois extensions""; ""Â1 Basic definitions and notation""; ""Â2 Hopf algebras and R-algebras""; ""Â3 Integrals""; ""Â4 Short exact sequences of Hopf algebras""; ""Â5 Hopf orders""; ""Chapter 2. Hopf Galois structures on separable field extensions""; ""Â6 Greither-Pareigis theory""; ""Â7 Byott's translation""; ""Â8 Byott's uniqueness theorem""; ""Â9 Prime power cyclic Galois extensions""; ""Â10 Non-abelian extensions""; ""Â11 Pure extensions""; ""Chapter 3. Tame extensions and Noether's Theorem""

""Â12 Associated orders""""Â13 Tame extensions""; ""Â14 Galois extensions""; ""Chapter 4. Hopf algebras of rank p""; ""Â15 Rank p Hopf orders over algebraically closed fields""; ""Â16 Eigenspace decompositions""; ""Chapter 5. Larson orders""; ""Â17 Group valuations""; ""Â18 Orders from valuations""; ""Â19 Lattice properties""; ""Â20 Uniqueness results""; ""Â21 One-parameter Larson orders""; ""Chapter 6. Cyclic extensions of degree p""; ""Â22 Different and discriminant""; ""Â23 Ramification groups""; ""Â24 Cyclic extensions of degree p""; ""Chapter 7. Non-maximal orders""

""Â25 Hopf Galois orders in Kummer extensions of order p""""Â26 Orders and associated orders""; ""Chapter 8. Ramification restrictions""; ""Â27 Ramification numbers""; ""Â28 Tameness""; ""Â29 When does tame imply Galois""; ""Â30 Ramification conditions""; ""Chapter 9. Hopf algebras of rank p[sup(2)]""; ""Â31 Hopf orders in KC[sub(p[sup(2)])]""; ""Chapter 10. Cyclic Hopf Galois extensions of degree p[sup(2)]""; ""Â32 Valuation rings of cyclic extensions of degree p[sup(2)]""; ""Chapter 11. Formal groups""; ""Â33 Formal groups""; ""Â34 Formal groups and Hopf algebras""

""Â35 Height""""Â36 Lubin-Tate formal groups""; ""Â37 Oort embedding""; ""Â38 Polynomial formal groups""; ""Chapter 12. Principal homogeneous spaces and formal groups""; ""Â39 Kummer extensions""; ""Â40 Which is the correct Hopf Galois structure""; ""Bibliography""; ""Index""; ""A""; ""B""; ""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""I""; ""K""; ""L""; ""M""; ""N""; ""O""; ""P""; ""R""; ""S""; ""T""; ""U""; ""V""; ""W""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 205-211) and index.

Description based on print version record.

ISBN:

1-4704-1307-8