*(English)*Zbl 1179.11040

Class field theory, the study of abelian extensions of local and global fields, is one of the major branches of algebraic number theory. Its roots can be traced back to the second half of the 19th century, when the search for possible generalizations of the quadratic and higher reciprocity laws of Gauss, Legendre, and others was initiated, mainly by the work of Eisenstein, Kummer, Kronecker, and Weber. In its modern abstract setting, class field theory was essentially developed in the first third of the 20th century, largely through the pioneering contributions by Hilbert, Takagi, E. Artin, Hasse, Furtwängler, and Grunwald.

The first comprehensive account of class field theory for number fields, reflecting its development until 1930, was provided by *H. Hasse’s* famous “Zahlbericht” (Zbl 0138.03202). In the following two decades, further decisive progress in class field theory was achieved by inventing new abstract concepts and tools such as group cohomology, idèle groups, Brauer groups, and class formations. These developments culminated in a highly general and powerful approach to class field theory, which was presented in the course of a seminar run by *E. Artin* (1898–1962) and *J. Tate* (born 1925) at Princeton University during the academic year 1951/1952.

The notes of this pioneering Artin-Tate seminar first circulated in mimeographed form, and were edited by Harvard University Press in 1961. Finally, in 1968, six years after E. Artin’s death, the main part of these notes appeared as the celebrated monograph “Class Field Theory” by *E. Artin* and *J. Tate* [New York-Amsterdam: W. A. Benjamin, Inc. (1968; Zbl 0176.33504)]. Together with *C. Chevalley’s* booklet “Class Field Theory” [Nagoya: Mathematical Institute, Nagoya University (1954; Zbl 0059.03304)], which had appeared in 1954 (Nagoya University Press), and with *J. Neukirch’s* first textbook on the subject (Zbl 0199.37502), the monograph by Artin and Tate was the standard reference in modern class field theory for the following decades.

Due to its undiminished significance and popularity, the classic “Class Field Theory” by *E. Artin* and *J. Tate* was reprinted in 1990 (Zbl 0681.12003), this time in the series “Advanced Book Classics” by Addison-Wesley Publishing Company.

The book under review is a new, slightly revised edition of this venerable classic of timeless importance and beauty, which certainly represents one of the great milestones in 20th century mathematics.

John Tate himself has enriched the text by a number of footnotes and historical comments, there by compensating for the lack of precise references and attribution of credit in the original edition. Moreover, and more importantly, he has added two substantial mathematical topics to the text. The first one is a sketch of the classic analytic proof of the second fundamental inequality in global class field theory, which uses zeta functions and Dirichlet densities. This provides an enlightening contrast to the algebraic proof via Kummer theory in Chapter VI.

The second addition concerns the final Chapter XV on Weil groups, where the existence and uniqueness of Weil groups for finite Galois extensions is now established in a general abstract way, basically by passing to a suitable inverse limit construction. Also, John Tale has written a new preface to the current edition, thereby explaining the contents from a more contemporary point of view, including references to the recent literature on the subject. In this vein, the bibliography has been updated accordingly. Finally, many typographical errors have been corrected, and the book appears now in modern $$ printing. Otherwise, both the original text and the numbering of the chapters have been left entirely intact, on account of which we may refer to the review of the original edition (Zbl 0176.33504) as for the precise contents of the book.

In fact, the pioneering spirit, the expository mastery, the inimitable elegance, and the fundamental significance of this great text in modern class field theory are still as vivid as they were over the past six decades. No doubt, this book will maintain its outstanding role in the field for many more decades to come, because it is simply matchless and irrecoverable as a source in modern algebraic number theory and its allied areas in contemporary mathematics as a whole.

##### MSC:

11R37 | Class field theory for global fields |

11R23 | Iwasawa theory |

11-02 | Research monographs (number theory) |

11-01 | Textbooks (number theory) |

01A75 | Collected or selected works |

11R34 | Galois cohomology for global fields |

11S15 | Ramification and extension theory |

11S31 | Class field theory for local fields; $p$-adic formal groups |