Galois Cohomology and Class Field Theory

Galois Cohomology and Class Field Theory
Author: David Harari
Publisher: Springer Nature
Total Pages: 336
Release: 2020-06-24
Genre: Mathematics
ISBN: 3030439011

This graduate textbook offers an introduction to modern methods in number theory. It gives a complete account of the main results of class field theory as well as the Poitou-Tate duality theorems, considered crowning achievements of modern number theory. Assuming a first graduate course in algebra and number theory, the book begins with an introduction to group and Galois cohomology. Local fields and local class field theory, including Lubin-Tate formal group laws, are covered next, followed by global class field theory and the description of abelian extensions of global fields. The final part of the book gives an accessible yet complete exposition of the Poitou-Tate duality theorems. Two appendices cover the necessary background in homological algebra and the analytic theory of Dirichlet L-series, including the Čebotarev density theorem. Based on several advanced courses given by the author, this textbook has been written for graduate students. Including complete proofs and numerous exercises, the book will also appeal to more experienced mathematicians, either as a text to learn the subject or as a reference.

A Gentle Course in Local Class Field Theory

A Gentle Course in Local Class Field Theory
Author: Pierre Guillot
Publisher: Cambridge University Press
Total Pages: 309
Release: 2018-11
Genre: Mathematics
ISBN: 1108421776

A self-contained exposition of local class field theory for students in advanced algebra.

Algebraic Groups and Class Fields

Algebraic Groups and Class Fields
Author: Jean-Pierre Serre
Publisher: Springer Science & Business Media
Total Pages: 220
Release: 2012-12-06
Genre: Mathematics
ISBN: 1461210356

Translation of the French Edition

Galois Theory of p-Extensions

Galois Theory of p-Extensions
Author: Helmut Koch
Publisher: Springer Science & Business Media
Total Pages: 196
Release: 2013-03-09
Genre: Mathematics
ISBN: 3662049678

Helmut Koch's classic is now available in English. Competently translated by Franz Lemmermeyer, it introduces the theory of pro-p groups and their cohomology. The book contains a postscript on the recent development of the field written by H. Koch and F. Lemmermeyer, along with many additional recent references.

Galois Cohomology

Galois Cohomology
Author: Jean-Pierre Serre
Publisher: Springer Science & Business Media
Total Pages: 215
Release: 2013-12-01
Genre: Mathematics
ISBN: 3642591418

This is an updated English translation of Cohomologie Galoisienne, published more than thirty years ago as one of the very first versions of Lecture Notes in Mathematics. It includes a reproduction of an influential paper by R. Steinberg, together with some new material and an expanded bibliography.

Class Field Theory

Class Field Theory
Author: J. Neukirch
Publisher: Springer Science & Business Media
Total Pages: 148
Release: 2012-12-06
Genre: Mathematics
ISBN: 364282465X

Class field theory, which is so immediately compelling in its main assertions, has, ever since its invention, suffered from the fact that its proofs have required a complicated and, by comparison with the results, rather imper spicuous system of arguments which have tended to jump around all over the place. My earlier presentation of the theory [41] has strengthened me in the belief that a highly elaborate mechanism, such as, for example, cohomol ogy, might not be adequate for a number-theoretical law admitting a very direct formulation, and that the truth of such a law must be susceptible to a far more immediate insight. I was determined to write the present, new account of class field theory by the discovery that, in fact, both the local and the global reciprocity laws may be subsumed under a purely group theoretical principle, admitting an entirely elementary description. This de scription makes possible a new foundation for the entire theory. The rapid advance to the main theorems of class field theory which results from this approach has made it possible to include in this volume the most important consequences and elaborations, and further related theories, with the excep tion of the cohomology version which I have this time excluded. This remains a significant variant, rich in application, but its principal results should be directly obtained from the material treated here.

Arithmetic Duality Theorems

Arithmetic Duality Theorems
Author: J. S. Milne
Publisher:
Total Pages: 440
Release: 1986
Genre: Mathematics
ISBN:

Here, published for the first time, are the complete proofs of the fundamental arithmetic duality theorems that have come to play an increasingly important role in number theory and arithmetic geometry. The text covers these theorems in Galois cohomology, ,tale cohomology, and flat cohomology and addresses applications in the above areas. The writing is expository and the book will serve as an invaluable reference text as well as an excellent introduction to the subject.

Cohomology of Number Fields

Cohomology of Number Fields
Author: Jürgen Neukirch
Publisher: Springer Science & Business Media
Total Pages: 831
Release: 2013-09-26
Genre: Mathematics
ISBN: 3540378898

This second edition is a corrected and extended version of the first. It is a textbook for students, as well as a reference book for the working mathematician, on cohomological topics in number theory. In all it is a virtually complete treatment of a vast array of central topics in algebraic number theory. New material is introduced here on duality theorems for unramified and tamely ramified extensions as well as a careful analysis of 2-extensions of real number fields.

Class Field Theory

Class Field Theory
Author: Georges Gras
Publisher: Springer Science & Business Media
Total Pages: 517
Release: 2013-11-11
Genre: Mathematics
ISBN: 3662113236

Global class field theory is a major achievement of algebraic number theory based on the functorial properties of the reciprocity map and the existence theorem. This book explores the consequences and the practical use of these results in detailed studies and illustrations of classical subjects. In the corrected second printing 2005, the author improves many details all through the book.