A Gentle Course in Local Class Field Theory

'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, Ecole normale superieure '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 'The title intrigues! How could anyone possibly introduce class field theory (local or global) gently? ... If one can't reasonably expect any author to anticipate and answer all the questions an expert teacher might field, Guillot comes as close as one might hope. Even theoretical courses need a goal, and this one culminates with the landmark Kronecker-Weber theorems, both local and global, characterizing all the abelian extensions of p-adic fields and of the rationals, respectively.' D. V. Feldman, Choice '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, Ecole normale superieure '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 'The title intrigues! How could anyone possibly introduce class field theory (local or global) gently? ... If one can't reasonably expect any author to anticipate and answer all the questions an expert teacher might field, Guillot comes as close as one might hope. Even theoretical courses need a goal, and this one culminates with the landmark Kronecker-Weber theorems, both local and global, characterizing all the abelian extensions of p-adic fields and of the rationals, respectively.' D. V. Feldman, Choice