Algebraic Patching

Assuming only basic algebra and Galois theory, the book develops the method of "e;algebraic patching"e; 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 "e;ample fields"e;. Among others, it leads to the solution of two central results in "e;Field Arithmetic"e;: (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 $C$.