group of a field and the character group of the absolute Galois group of the algebraic closure of the field is proved. Ramification theory for abelian extensions is developed. Chapter XIII The global class field theory for A-fields is developed using the pairings of Chapter XII, replacing multiplicative groups of local fields with idèle class groups of A-fields. The pairing is constructed as a product over places of local Hasse invariants. Third edition Some references are added, some minor corrections made, some comments added, and five appendices are included, containing the following material: A character version of the (local) transfer theorem and its extension to the global transfer theorem. Šafarevič's theorem on the structure of Galois groups of local fields using the theory of Weil groups. Theorems of Tate and Sen on the Herbrand distribution. Examples of L-functions with Grössencharacter. Editions References Mathematics books Algebraic number theory