Groups as Galois Groups An Introduction [Paperback]

Develops the mathematical background and recent results on the Inverse Galois Problem.This book is on the theory of symmetries in solutions of algebraic equations. It describes the Inverse Galois Problem, a classical unsolved problem posed by Hilbert at the beginning of the century, which brings together ideas from group theory, algebraic geometry and number theory, topology, and analysis. The author begins from the foundations and develops the necessary mathematical background to lead the reader to the research frontier. Graduate students and mathematicians from other areas will find this an excellent introduction to a fascinating field.This book is on the theory of symmetries in solutions of algebraic equations. It describes the Inverse Galois Problem, a classical unsolved problem posed by Hilbert at the beginning of the century, which brings together ideas from group theory, algebraic geometry and number theory, topology, and analysis. The author begins from the foundations and develops the necessary mathematical background to lead the reader to the research frontier. Graduate students and mathematicians from other areas will find this an excellent introduction to a fascinating field.This book describes various approaches to the Inverse Galois Problem, a classical unsolved problem of mathematics posed by Hilbert at the beginning of the century. It brings together ideas from group theory, algebraic geometry and number theory, topology, and analysis. Assuming only elementary algebra and complex analysis, the author develops the necessary background from topology, Riemann surface theory and number theory. The first part of the book is quite elementary, and leads up to the basic rigidity criteria for the realization of groups as Galois groups. The second part presents more advanced topics, such as braid group action and moduli spaces for covers of the Riemann sphere, GAR- and GAL- realizations, and patching over complete valued fields. Graduate students and mathl.