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

1 | |

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 |

205 | |

213 | |

### Other editions - View all

Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory: Hopf ... Lindsay Childs No preview available - 2000 |

### Common terms and phrases

abelian action acts apply associated order assume basis Chapter classification cocommutative commutative consider construction contains COROLLARY corresponding cyclic of order defined DEFINITION dimension divides dual element embedding equivalent example extension L/K finite follows formal group Galois extension Galois group given group G H & H H-Galois H-module hence Hol(N homomorphism Hopf algebra Hopf Galois structures Hopf order ideal identity implies induced integrals of H isogeny isomorphism Kummer Larson order LEMMA Let G Let H Let L/K maximal module multiplication normal obtain order in KG parameter Perm(X polynomial prime primitive projective PROOF properties PROPOSITION R-algebra R-Hopf ramification number rank Recall respectively result satisfies sequence Suppose theorem theory totally ramified unique unit valuation ring yields