1. Algebraic Patching -- 2. Normed Rings -- 3. Several Variables -- 4. Constant Split Embedding Problems over Complete Fields -- 5. Ample Fields -- 6. Non-Ample Fields -- 7. Split Embedding Problems over Complete Fields -- 8. Split Embedding Problems over Ample Fields -- 9. The Absolute Galois Group of C(t) -- 10. Semi-Free Profinite Groups -- 11. Function Fields of One Variable over PAC Fields -- 12. Complete Noetherian Domains -- Open Problems -- References -- Glossary of Notation -- Index.
Assuming only basic algebra and Galois theory, the book develops the method of "algebraic patching" to realize finite groups and, more generally, to solve finite split embedding problems over fields. The method succeeds over rational function fields of one variable over "ample fields". Among others, it leads to the solution of two central results in "Field Arithmetic": (a) The absolute Galois group of a countable Hilbertian pac field is free on countably many generators; (b) The absolute Galois group of a function field of one variable over an algebraically closed field $C$ is free of rank equal to the cardinality of $C$.