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Introduction to the second edition | ||
Foreword to the first edition | ||
Ch. I | Complete Discrete Valuation Fields | 1 |
1 | Ultrametric Absolute Values | 1 |
2 | Valuations and Valuation Fields | 4 |
3 | Discrete Valuation Fields | 6 |
4 | Completion | 9 |
5 | Filtrations of Discrete Valuation Fields | 12 |
6 | The Group of Principal Units as a Z[subscript p]-module | 17 |
7 | Set of Multiplicative Representatives | 22 |
8 | The Witt ring | 26 |
9 | Artin-Hasse Maps | 29 |
Ch. II | Extensions of Discrete Valuation Fields | 35 |
1 | The Hensel Lemma and Henselian Fields | 35 |
2 | Extensions of Valuation Fields | 39 |
3 | Unramified and Ramified Extensions | 49 |
4 | Galois Extensions | 56 |
5 | Structure Theorems for Complete Fields | 61 |
Ch. III | The Norm Map | 67 |
1 | Cyclic Extensions of Prime Degree | 67 |
2 | Artin-Schreier Extensions | 74 |
3 | The Hasse-Herbrand Function | 79 |
4 | The Norm and Ramification Groups | 88 |
5 | The Field of Norms | 95 |
Ch. IV | Local Class Field Theory. I | 111 |
1 | Useful Results on Local Fields | 112 |
2 | The Neukirch Map | 123 |
3 | The Hazewinkel Homomorphism | 128 |
4 | The Reciprocity Map | 139 |
5 | Pairings of the Multiplicative Group | 142 |
6 | The Existence Theorem | 153 |
7 | Other Approaches to the Local Reciprocity Map | 161 |
8 | Nonabelian Extensions | 165 |
Ch. V | Local Class Field Theory. II | 171 |
1 | The Multiplicative Group and Abelian Extensions | 171 |
2 | Additive Polynomials | 179 |
3 | Normic Subgroups | 187 |
4 | Local p-Class Field Theory | 196 |
5 | Generalizations | 203 |
Ch. VI | The Group of Units of Local Number Fields | 207 |
1 | Formal Power Series | 207 |
2 | The Artin-Hasse-Shafarevich Map | 214 |
3 | Series Associated to Roots | 219 |
4 | Primary Elements | 228 |
5 | The Shafarevich Basis | 232 |
Ch. VII | Explicit Formulas for the Hilbert Symbol | 235 |
1 | Origin of Formulas | 235 |
2 | The Pairing (.,.) | 241 |
3 | Explicit Class Field Theory for Kummer Extensions | 250 |
4 | Explicit Formulas | 255 |
5 | Applications and Generalizations | 258 |
Ch. VIII | Explicit Formulas for Hilbert Pairings on Formal Groups | 267 |
1 | Formal Groups | 267 |
2 | Generalized Hilbert Pairing for Lubin-Tate Groups | 272 |
3 | Generalized Hilbert Pairing for Honda Groups | 276 |
Ch. IX | The Milnor K-groups of a Local Field | 283 |
1 | The Milnor Ring of a Field | 283 |
2 | The Milnor Ring of a Discrete Valuation Field | 286 |
3 | The Norm Map | 293 |
4 | The Milnor Ring of a Local Field | 304 |
Bibliography | 319 | |
List of Notations | 341 | |
Index | 343 |
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Add Local fields and their extensions, This book offers a modern presentation of local fields whose spectacular development was initiated almost one hundred years ago by K. Hensel. The volume consists of nine chapters divided into four parts: arithmetic properties of local fields, class field , Local fields and their extensions to the inventory that you are selling on WonderClubX
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Add Local fields and their extensions, This book offers a modern presentation of local fields whose spectacular development was initiated almost one hundred years ago by K. Hensel. The volume consists of nine chapters divided into four parts: arithmetic properties of local fields, class field , Local fields and their extensions to your collection on WonderClub |