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**Local fields and their extensions.
2nd ed.**
*(English)*
Zbl 1156.11046

Translations of Mathematical Monographs 121. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3259-X/hbk). xi, 345 p. (2002).

It has been ten years since the publication of the first edition, therefore this edition incorporates many improvements to the first edition, with 60 additional pages reflecting several aspects of the developments in local field theory. For the review of the original (1993) see Zbl 0781.11042.

From the text: “The volume consists of four parts: elementary properties of local fields, class field theory for various types of local fields and generalizations, explicit formulas for the Hilbert pairing, and Milnor \(K\)-groups of fields and of local fields. The first three parts essentially simplify, revise, and update the first edition. The book includes the following recent topics: Fontaine-Wintenberger theory of arithmetically profinite extensions and fields of norms, explicit noncohomological approach to the reciprocity map with a review of all other approaches to local class field theory, Fesenko’s \(p\)-class field theory for local fields with perfect residue field, simplified updated presentation of Vostokov’s explicit formulas for the Hilbert norm residue symbol, and Milnor \(K\)-groups of local fields. Numerous exercises introduce the reader to other important recent results in local number theory, and an extensive bibliography provides a guide to related areas.”

The book is a ‘must’ for graduate students and research mathematicians interested in local field theory and its applications in arithmetic algebraic geometry.

From the text: “The volume consists of four parts: elementary properties of local fields, class field theory for various types of local fields and generalizations, explicit formulas for the Hilbert pairing, and Milnor \(K\)-groups of fields and of local fields. The first three parts essentially simplify, revise, and update the first edition. The book includes the following recent topics: Fontaine-Wintenberger theory of arithmetically profinite extensions and fields of norms, explicit noncohomological approach to the reciprocity map with a review of all other approaches to local class field theory, Fesenko’s \(p\)-class field theory for local fields with perfect residue field, simplified updated presentation of Vostokov’s explicit formulas for the Hilbert norm residue symbol, and Milnor \(K\)-groups of local fields. Numerous exercises introduce the reader to other important recent results in local number theory, and an extensive bibliography provides a guide to related areas.”

The book is a ‘must’ for graduate students and research mathematicians interested in local field theory and its applications in arithmetic algebraic geometry.

Reviewer: Olaf Ninnemann (Berlin)

### MSC:

11Sxx | Algebraic number theory: local fields |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11S31 | Class field theory; \(p\)-adic formal groups |

11S15 | Ramification and extension theory |

11S70 | \(K\)-theory of local fields |

11S99 | Algebraic number theory: local fields |

11S20 | Galois theory |

14G20 | Local ground fields in algebraic geometry |