Cohomology of Number Fields (Grundlehren der mathematischen Wissenschaften, 323)

From the reviews of the second edition: "The publication of a second edition gives me a chance to ... emphasize what an important book it is. ... the book a necessary part of the number theorist’s library. That it’s also well written, clear, and systematic is a very welcome bonus. ... There are many goodies here ... . it is an indispensable book for anyone working in number theory. ... Neukirch, Schmidt, and Wingberg have, in fact, produced ... authoritative, complete, careful, and sure to be a reliable reference for many years." (Fernando Q. Gouvêa, MathDL, May, 2008) "The second edition will continue to serve as a very helpful and up-to-date reference in cohomology of profinite groups and algebraic number theory, and all the additions are interesting and useful. ... the book is fine as it is: systematic, very comprehensive, and well-organised. This second edition will be a standard reference from the outset, continuing the success of the first one." (Cornelius Greither, Zentralblatt MATH, Vol. 1136 (14), 2008) The 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. The First Part Provides Algebraic Background: Cohomology Of Profinite Groups, Duality Groups, Free Products, And Homotopy Theory Of Modules, With New Sections On Spectral Sequences And On Tate Cohomology Of Profinite Groups. The Second Part Deals With Galois Groups Of Local And Global Fields: Tate Duality, Structure Of Absolute Galois Groups Of Local Fields, Extensions With Restricted Ramification, Poitou-tate Duality, Hasse Principles, Theorem Of Grunwald-wang, Leopoldt’s Conjecture, Riemann’s Existence Theorem, The Theorems Of Iwasawa And Of Šafarevic On Solvable Groups As Galois Groups, Iwasawa Theory, And Anabelian Principles. New Material Is Introduced Here On Duality Theorems For Unramified And Tamely Ramified Extensions, A Careful Analysis Of 2-extensions Of Real Number Fields And A Complete Proof Of Neukirch’s Theorem On Solvable Galois Groups With Given Local Conditions. The Present Edition Is A Corrected Printing Of The 2008 Edition. Part I Algebraic Theory: Cohomology Of Profinite Groups -- Some Homological Algebra -- Duality Properties Of Profinite Groups -- Free Products Of Profinite Groups -- Iwasawa Modules -- Part Ii Arithmetic Theory: Galois Cohomology -- Cohomology Of Local Fields -- Cohomology Of Global Fields -- The Absolute Galois Group Of A Global Field -- Restricted Ramification -- Iwasawa Theory Of Number Fields -- Anabelian Geometry -- Literature -- Index. Jürgen Neukirch, Alexander Schmidt, Kay Wingberg. Includes Bibliographical References (p. [805]-819) And Index. The 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. The first part provides algebraic background : cohomology of profinite groups, duality groups, free products, and homotopy theory of modules, with new sections on spectral sequences and on Tate cohomology of profinite groups. The second part deals with Galois groups of local and global fields : Tate duality, structure of absolute Galois groups of local fields, extensions with restricted ramification, Poitou-Tate duality, Hasse principles, theorem of Grunwald-Wang, Leopoldts conjecture, Riemanns existence theorem, the theorems of Iwasawa and of afarevic on solvable groups as Galois groups, Iwasawa theory, and anabelian principles. New material is introduced here on duality theorems for unramified and tamely ramified extensions, a careful analysis of 2-extensions of real number fields and a complete proof of Neukirchs theorem on solvable Galois groups with given local conditions. The present edition is a corrected printing of the 2008 edition. -- [Source inconnue] Galois modules over local and global fields form the main subject of this monograph, which can serve both as a textbook for students, and as a reference book for the working mathematician, on cohomological topics in number theory. The first part provides the necessary algebraic background. The arithmetic part deals with Galois groups of local and global local Tate duality, the structure of the absolute Galois group of a local field, extensions of global fields with restricted ramification, cohomology of the idle and the idle class groups, Poitou-Tate duality for finitely generated Galois modules, the Hasse principle, the theorem of Grunwald-Wang, Leopoldt's conjecture, Riemann's existence theorem for number fields, embedding problems, the theorems of Iwasawa and of Safarevic on solvable groups as Galois groups over global fields, Iwasawa theory of local and global number fields, and the characterization of number fields by their absolute Galois groups. Front Matter....Pages i-xv Front Matter....Pages 1-1 Cohomology of Profinite Groups....Pages 3-96 Some Homological Algebra....Pages 97-145 Duality Properties of Profinite Groups....Pages 147-244 Free Products of Profinite Groups....Pages 245-266 Iwasawa Modules....Pages 267-333 Front Matter....Pages 335-335 Galois Cohomology....Pages 337-369 Cohomology of Local Fields....Pages 371-423 Cohomology of Global Fields....Pages 425-520 The Absolute Galois Group of a Global Field....Pages 521-597 Restricted Ramification....Pages 599-719 Iwasawa Theory of Number Fields....Pages 721-784 Anabelian Geometry....Pages 785-803 Back Matter....Pages 805-826 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. Focuses on Galois modules over local and global fields. This book covers the following topics: local Tate duality, the structure of the absolute Galois group of a local field, extensions of global fields with restricted ramification, cohomology of the idele and the idele class groups, and more. Profinite groups are topological groups which naturally occur in algebraic number theory as Galois groups of infinite field extensions or more generally as étale fundamental groups of schemes.