United States in 1967, the author chooses to drive this particular viewpoint home by explaining that the two classes of global fields “must be granted a fully simultaneous treatment […] instead of the segregated status, and at best the separate but equal facilities, which hitherto have been their lot. That, far from losing by such treatment, both races stand to gain by it, is one fact which will, I hope, clearly emerge from this book.” After World War II, a series of developments in class field theory diminished the significance of the cyclic algebras (and, more generally, the crossed product algebras) which are defined in terms of the number field in proofs of class field theory. Instead cohomological formalism became a more significant part of local and global class field theory, particularly in work of Hochschild and Nakayama, Weil, Artin, and Tate during the period 1950–1952. Alongside the desire to consider algebraic number fields alongside function fields over finite fields, the work of Chevalley is particularly emphasised. In order to derive the theorems of global class field theory from those of local class field theory, Chevalley introduced what he called the élément idéal, later called idèle, at Hasse's suggestion. The idèle group of a number field was first introduced by Chevalley in order to describe global class field theory for infinite extensions, but several years later he used it in a new way to derive global class field theory from local class field theory. Weil mentioned this (unpublished) work as a significant influence on some of the choices of treatment he uses. Reception The 1st edition was reviewed by George Whaples for Mathematical Reviews and Helmut Koch for Zentralblatt. Later editions were reviewed by Fernando Q. Gouvêa for the Mathematical Association of America and by W. Zink and Helmut Koch for Zentralblatt; in his review of the second edition Koch makes the remark "Shafarevich showed me the first edition in autumn 1967 in Moscow and said that this book will be from now on the book about class field theory". The coherence of the treatment, and some of its distinctive features, were highlighted by several reviewers, with Koch going on to say "This book is written in the spirit of the early forties and just this makes it a valuable source of information for everyone who is working about problems related to number and function fields." Contents Roughly speaking, the