This volume is an English translation of "Cohomologie Galoisienne" . The original edition (Springer LN5, 1964) was based on the notes, written with the help of Michel Raynaud, of a course I gave at the College de France in 1962-1963. In the present edition there are numerous additions and one suppression: Verdier's text on the duality of profinite groups. The most important addition is the photographic reproduction of R. Steinberg's "Regular elements of semisimple algebraic groups", Publ. Math. LH.E.S., 1965. I am very grateful to him, and to LH.E.S., for having authorized this reproduction. Other additions include: - A proof of the Golod-Shafarevich inequality (Chap. I, App. 2). - The "resume de cours" of my 1991-1992 lectures at the College de France on Galois cohomology of k(T) (Chap. II, App.). - The "resume de cours" of my 1990-1991 lectures at the College de France on Galois cohomology of semisimple groups, and its relation with abelian cohomology, especially in dimension 3 (Chap. III, App. 2). The bibliography has been extended, open questions have been updated (as far as possible) and several exercises have been added. In order to facilitate references, the numbering of propositions, lemmas and theorems has been kept as in the original 1964 text. Jean-Pierre Serre Harvard, Fall 1996 Table of Contents Foreword ........................................................ V Chapter I. Cohomology of profinite groups ยง1. Profinite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . .
I. Cohomology of profinite groups -- ยง1. Profinite groups -- ยง2. Cohomology -- ยง3. Cohomological dimension -- ยง4. Cohomology of pro-p-groups -- ยง5. Nonabelian cohomology -- II. Galois cohomology, the commutative case -- ยง1. Generalities -- ยง2. Criteria for cohomological dimension -- ยง3. Fields of dimension ?1 -- ยง4. Transition theorems -- ยง5. p-adic fields -- ยง6. Algebraic number fields -- III. Nonabelian Galois cohomology -- ยง1. Forms -- ยง2. Fields of dimension ? 1 -- ยง3. Fields of dimension ? 2 -- ยง4. Finiteness theorems