This book is devoted to the structure of the absolute Galois groups of certain algebraic extensions of the field of rational numbers. Its main result, a theorem proved by the authors and Florian Pop in 2012, describes the absolute Galois group of distinguished semi-local algebraic (and other) extensions of the rational numbers as free products of the free profinite group on countably many generators and local Galois groups. This is an instance of a positive answer to the generalized inverse problem of Galois theory.

Adopting both an arithmetic and probabilistic approach, the book carefully sets out the preliminary material needed to prove the main theorem and its supporting results. In addition, it includes a description of Melnikov's construction of free products of profinite groups and, for the first time in book form, an account of a generalization of the theory of free products of profinite groups and their subgroups.

The book will be of interest to researchers in field arithmetic, Galois theory and profinite groups.

About the Author

Moshe Jarden is professor emeritus at the School of Mathematics, Tel Aviv University. He obtained his PhD at the Hebrew University, Jerusalem, in 1970, and his Habilitation at Heidelberg University in 1972, where he was a post doc until 1974, before joining Tel Aviv University. He was awarded the L. Meitner–A.v. Humboldt prize in 2001 for his achievements in mathematics. He is the author of two books, Field Arithmetic (for which he was awarded the Landau Prize in 1987) and Algebraic Patching, and he has published 120 research articles. His research is primarily on field arithmetic.

Dan Haran is a professor at the School of Mathematics, Tel Aviv University, where he obtained his PhD in 1983. He was a research fellow at the Mathematical Institute in Erlangen (1983–1986), a visiting assistant professor at Rutgers (1985–1986), a senior lecturer at Tel Aviv University (1986–1991), and a fellow at the Hebrew University (1991–1992) and at the Max-Planck-Institut,Bonn (1992–1993) before joining Tel Aviv University in 1991 as an associate professor, becoming full professor in 2000. His research is in field arithmetic, Galois theory and profinite groups.

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