The Absolute Galois Group of a Semi-Local Field (Springer Monographs in Mathematics)

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. Introduction Contents Notation and Conventions Chapter 1 Topologies 1.1 Profinite Spaces 1.2 Strict and Étale Topologies 1.3 Étale Hausdorff Sets 1.4 The Envelope of a Set of Closed Subgroups 1.5 Cantor Space Chapter 2 Families of Subgroups 2.1 Continuous and Separated Families 2.2 Sheaves of Profinite Groups 2.3 Sheaf-Group Structures Chapter 3 Free Products of Finitely Many Profinite Groups 3.1 Definition and First Properties 3.2 Finite Subgroups of Free Products of Finitely Many Profinite Groups Chapter 4 Generalized Free Products 4.1 Inner Free Products 4.2 Outer Free Products 4.3 Equivalence of Inner and Outer Free Products 4.4 Free Profinite Products in the Sense of Binz–Neukirch–Wenzel 4.5 Properties of Inner Free Products 4.6 Further Properties of Inner Free Products 4.7 Free Products as Semi-direct Products 4.8 Closed Subgroups of Free Products Chapter 5 Relative Embedding Problems 5.1 Embedding Problems and Projectivity 5.2 Strongly G-Projective Groups 5.3 Infinite Embedding Problems 5.4 The Closed Subgroup Theorem 5.5 The Set G_max 5.6 Prescribed Solutions Chapter 6 Strong Proper Projectivity 6.1 Prosolvable Subgroups of G-Projective Groups 6.2 Groups of Local p-Type 6.3 Fields of Local p-Type 6.4 Groups of P-Type and G-Projectivity Chapter 7 A Free Profinite Product over an Étale Compact Subset of Subgr(G) 7.1 A System of Representatives 7.2 The Generalized Iwasawa Theorem Chapter 8 Fundamental Result 8.1 Classical Closures of a Field 8.2 Sections over Henselian Fields 8.3 G-Projectivity 8.4 Semi-Local Fields Chapter 9 Main Result 9.1 The Groups Gal(K_tot,s[σ]) 9.2 Finitely Generated Groups References Symbol Index Subject Index