## A Gentle Course in Local Class Field Theory: Local Number Fields, Brauer Groups, Galois CohomologyThis book offers a self-contained exposition of local class field theory, serving as a second course on Galois theory. It opens with a discussion of several fundamental topics in algebra, such as profinite groups, p-adic fields, semisimple algebras and their modules, and homological algebra with the example of group cohomology. The book culminates with the description of the abelian extensions of local number fields, as well as the celebrated Kronecker-Weber theory, in both the local and global cases. The material will find use across disciplines, including number theory, representation theory, algebraic geometry, and algebraic topology. Written for beginning graduate students and advanced undergraduates, this book can be used in the classroom or for independent study. |

### Contents

Local number fields | 23 |

Around Hensels lemma | 31 |

Unramified extensions | 43 |

Higher ramification groups | 49 |

Infinite Galois extensions | 62 |

The multiplicative structure of local number fields | 72 |

BRAUER GROUPS | 83 |

5 | 85 |

8 | 146 |

GALOIS COHOMOLOGY | 157 |

10 | 187 |

Hilbert 90 | 204 |

Finer structure | 214 |

CLASS FIELD THEORY | 241 |

14 | 261 |

background material | 281 |

### Other editions - View all

A Gentle Course in Local Class Field Theory: Local Number Fields, Brauer ... Pierre Guillot No preview available - 2018 |

### Common terms and phrases

A-module abelian extension abelian group algebraic closure assume bijection Br(F Brauer equivalent Brauer group called chain complex chapter class field theory cochain cocycle coefficients cohomology groups consider contained corestriction Corollary cup-products cyclic group deduce define definition degree dimension direct sum element End A(V Example exists extension L/F F-algebra fact field F finite and Galois finite group follows Gal(K/F given group G homology homotopy Homr identity injective integer inverse isomorphism kernel Kummer Let F Let G long exact sequence matrix Mn(F morphism multiplication nonzero norm notation nth root number field p-adic fields phism polynomial prime number profinite group projective resolution Proof Proposition prove reader result ring root of unity semisimple short exact sequence simple F-algebra simple modules skewfield splitting field subgroup submodule Suppose surjective theorem topology trivial unique unramified valuation vector space write Z[G]-module Z/nZ zig-zag lemma