Berlin: Springer. x, 210 pp. DM 78.00; öS 569.40; sFr. 69.00; £32.00; $ 59.95 (1997).

This is a translation into English of the revised and completed fifth edition of Serre’s “Cohomologie Galoisienne”. It is a close and faithful translation and contains all of the added material of the latest edition. This English edition also contains a photographic reproduction of the important paper, ‘Regular elements of semisimple algebraic groups’ [Inst. Hautes Etud. Sci., Publ. Math. 25, 281-312 (1965;

Zbl 0136.30002)] by

*R. Steinberg*, in which the following conjecture of Serre is proved: If

$k$ is a perfect field of cohomological dimension at most 1 and if

$G$ is a connected linear algebraic group defined over

$k$, then

${H}^{1}(k,G)=0$.