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Look Inside A Gentle Course in Local Class Field Theory

A Gentle Course in Local Class Field Theory
Local Number Fields, Brauer Groups, Galois Cohomology

$39.99 (P)

  • Publication planned for: October 2018
  • availability: Not yet published - available from October 2018
  • format: Paperback
  • isbn: 9781108432245

$ 39.99 (P)
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  • This 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.

    • Written for students rather than experts by integrating exposition and results
    • Takes a coherent, self-contained path to the fundamental theorem of class field theory
    • Teaches skills that are applicable in multiple contexts
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    Reviews & endorsements

    Advance praise: 'This masterly written introductory course in number theory and Galois cohomology fills a gap in the literature. Readers will find a complete and nevertheless very accessible treatment of local class field theory and, along the way, comprehensive introductions to topics of independent interest such as Brauer groups or Galois cohomology. Pierre Guillot's book succeeds at presenting these topics in remarkable depth while avoiding the pitfalls of maximal generality. Undoubtedly a precious resource for students of Galois theory.' Olivier Wittenberg, École normale supérieure

    Advance praise: 'Class field theory, and the ingredients of its proofs (e.g. Galois Cohomology and Brauer groups), are cornerstones of modern algebra and number theory. This excellent book provides a clear introduction, with a very thorough treatment of background material and an abundance of exercises. This is an exciting and indispensable book to anyone who works in this field.' David Zureick-Brown, Emory University, Georgia

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    Product details

    • Publication planned for: October 2018
    • format: Paperback
    • isbn: 9781108432245
    • dimensions: 247 x 174 mm
    • availability: Not yet published - available from October 2018
  • Table of Contents

    Part I. Preliminaries:
    1. Kummer theory
    2. Local number fields
    3. Tools from topology
    4. The multiplicative structure of local number fields
    Part II. Brauer Groups:
    5. Skewfields, algebras, and modules
    6. Central simple algebras
    7. Combinatorial constructions
    8. The Brauer group of a local number field
    Part III. Galois Cohomology:
    9. Ext and Tor
    10. Group cohomology
    11. Hilbert 90
    12. Finer structure
    Part IV. Class Field Theory:
    13. Local class field theory
    14. An introduction to number fields.

  • Author

    Pierre Guillot, Université de Strasbourg
    Pierre Guillot is a lecturer at the Université de Strasbourg and a researcher at the Institut de Recherche Mathématique Avancée (IRMA). He has authored numerous research papers in the areas of algebraic geometry, algebraic topology, quantum algebra, knot theory, combinatorics, the theory of Grothendieck's dessins d'enfants, and Galois cohomology.

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