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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 $C$.
Inhaltsverzeichnis
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.
Über den Autor / die Autorin
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.
Zusammenfassung
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 $C$.
