*(English)*Zbl 1055.12003

Since the appearance of the first edition of *Field Arithmetic* in 1986, the research on the subject has remarkably increased. The goal of this new edition is to enrich the book with an extensive account of the progress made in this field, even if, as the authors say, the task of giving a fully complete account in just one book is probably beyond reach.

In this review, also necessarily incomplete, we recall a few of the major changes introduced in the new edition, referring to Zbl 0625.12001 for comments to the first edition.

The background material, roughly speaking the material contained in Chapters 1 to 12, has been reorganized and enlarged; in particular, arguments like linear disjointness of fields and algebraic function fields of one variable have been significantly expanded. Chapter 13, devoted to classical Hilbertian fields, now includes recent results of Zannier and Haran, which lead to the solution of classical problems about the Hilbertianity of some fields (see the list of open problems in the first edition). Matzat’s results on the GAR realization of simple finite groups and their implications on the solvability of the embedding problem are included in Chapter 16. Melnikov’s formations $\mathcal{C}$, i.e., the sets of all finite groups whose composition factors belong to a given set of finite simple group, are discussed in Chapter 17. Chapter 21 includes a full proof of Schur’s conjecture about polynomials with coefficients in the ring of integers ${\mathcal{O}}_{K}$ of a number field $K$ that induce bijections on ${\mathcal{O}}_{k}/P$ for infinitely many prime ideals of ${\mathcal{O}}_{k}$. The study of free profinite groups of infinite rank has been substantially expanded in Chapter 25. Probabilistic arguments regarding generators of free profinite groups are given in Chapters 26.

As in the first edition, an important role to the development of the theory is recognized to the methods coming from logic. Therefore there are a number of chapters devoted to logical arguments, like ultraproducts, decision procedures, nonstandard model theory and undecidability.

Each chapter includes notes on related literature and almost every chapter has a list of exercises. The last chapter recalls the list of open problems of the first edition, with comments on partial or full solutions, and presents a new list of open problems.

In the reviewer’s opinion, the book is a very rich survey of results in Field Arithmetic and could be very heplful for specialists. On the other hand, it also contains a large number of results of independent interest, and therefore it is highly recommendable to many others too.