Algebraic Patching [Paperback]

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 $C$.

This 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 .

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.Moshe Jarden was born in 1942 in Tel Aviv, Israel. In 1970 hereceived his Ph.D in Mathematics from the Hebrew University ofJerusalem having Hillel Furstenberg as his thesis advisor. He spentthree years (1972-1974) at the Mathematisches Institut of Heidelbergwith Peter Roquette as his mentor and habilitated there in 1972. Theyears he spent in Heidelberg laid the foundation to a long termedcooperation with German mathematicians, especially with PeterRoquette, Wulf-Dieter Geyer, Gerhard Frey, and Juergen Ritter. As atoken to his achievements in Mathematics and his fruitfulcooperation with German mathematicians the AlexanderlãD