Cohomology of Number Fields (Grundlehren der mathematischen Wissenschaften)

I Algebraic Theory: Cohomology of Profinite Groups \* Some Homological Algebra \* Duality Properties of Profinite Groups \* Free Products of Profinite Groups \* Iwasawa Modules 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 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. 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.