435 pp., $109.00, ISBN 3-540-66986-8

The theory of profinite groups is flourishing! This is the first immediate observation from looking at the four books on the subject which have come out in the last two years. The subject, which only two decades ago was somewhat remote, has made its way to mainstream mathematics in several different ways. What is a profinite group? A profinite group G is a topological group which is Hausdorff, compact and totally disconnected. A more meaningful equivalent definition is that G is the inverse limit of finite groups. As such, much of the theory of finite groups can be carried over to profinite groups. The origin of the subject lies in Galois theory: If E is an infinite algebraic Galois extension of a field F and G = Gal(E/F ) is the group of the automorphisms of E fixing F , then the classical Galois theory does not hold anymore. Intermediate fields between F and E give rise to subgroups of G, but the fixed point sets of different subgroups of G can give the same intermediate subfield. Now G has a natural topology called the Krull topology obtained by declaring, for every finite subextension F ⊆ L ⊆ E, the subgroup G(E/L) to be open in G. The classical Galois theorem now holds in the following form: There is a one to one correspondence between intermediate field extensions and closed subgroups of G. So infinite Galois theory leads one naturally to profinite groups. Many questions of arithmetic interest are naturally formulated in this way. For example, the inverse Galois problem asking whether every finite group is a Galois group of some (finite) Galois extension of Q is actually the question whether every finite group is a quotient of the profinite group G(Q̄/Q) when Q̄ is the field of all algebraic numbers. Indeed, the profinite group G(Q̄/Q) has been the focus of a lot of research. More generally, “Field Arithmetic” (see [FJ]) is the area which studies these infinite field extensions, their Galois theory and model theoretic aspects. Profinite groups have come up also from a different direction. The notion of a Lie group is fundamental in many areas of mathematics: This is a topological group which is at the same time an analytic manifold, i.e., locally homeomorphic to a ball in R, so that the group operations are analytic functions in the local coordinates. One can replace R by any locally compact field over which “analytic functions” can be defined in particular by Qp, the field of p-adic numbers to get a “p-adic Lie group”. It turns out that such a p-adic Lie group always has an open