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Recent Developments in the Inverse Galois Problem
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Contemporary Mathematics
1995; 401 pp; softcover
Volume: 186
ISBN-10: 0-8218-0299-2
ISBN-13: 978-0-8218-0299-1
List Price: US$83 Member Price: US$66.40
Order Code: CONM/186

This book contains the refereed proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Recent Developments in the Inverse Galois Problem, held in July 1993 at the University of Washington, Seattle. A new review of Serre's Topics in Galois Theory serves as a starting point. The book describes the latest research on explicit presentation of the absolute Galois group of the rationals. Containing the first appearance of generalizations of modular curves, the book presents applications that demonstrate the full scope of the Inverse Galois Problem. In particular, the papers collected here show the ubiquity of the applications of the Inverse Galois Problem and its compelling significance. The book will serve as a guide to progress on the Inverse Galois Problem and as an aid in using this work in other areas of mathematics. This includes coding theory and other finite field applications. Group theory and a first course in algebraic curves are sufficient for understanding many papers in the volume. Graduate students will find this an excellent reference to current research, as it contains a list of problems appropriate for thesis material in arithmetic geometry, algebraic number theory, and group theory.

Researchers in number theory and group theory and students seeking applications of pure mathematics as down-to-earth problems.

Part A. Explicit quotients of $$G_{\mathbb Q}$$ and $$G_{\bar {\mathbb F}(t)}$$
• T. Crespo -- Explicit Galois realization of $$C_{16}$$-extensions of $$A_n$$ and $$S_n$$
• M. D. Fried -- Topics in Galois theory
• B. H. Matzat -- Parametric solutions of embedding problems
• A. Reverter and N. Vila -- Some projective linear groups over finite fields as Galois groups over $$\mathbb Q$$
• S. Liedahl and J. Sonn -- $$K$$-Admissibility of metacyclic $$2$$-groups
• J. R. Swallow -- Embedding problems and the $$C_{16}\rightarrow C_8$$ obstruction
• H. Völklein -- Cyclic covers of $$\mathbb P^1$$ and Galois action on their division points
Part B. Moduli spaces and the structure of $$G_{\mathbb Q}$$
• M. D. Fried -- Introduction to modular towers: Generalizing dihedral group-modular curve connections
• Y. Ihara and M. Matsumoto -- On Galois actions on profinite completions of braid groups
• M. Matsumoto -- On the Galois image in the derivation algebra of $$\pi _1$$ of the projective line minus three points
Part C. The structure of $$G_{\mathbb R(t)}, G_{\bar {\mathbb F}_q(t)}$$, and $$G_{\mathbb Q_p(t)}$$
• P. Dèbes -- Covers of $$\mathbb P^1$$ over the $$p$$-adics
• B. Deschamps -- Existence de points $$p$$-adiques pour tout $$p$$ sur un espace de Hurwitz
• E. Dew -- Stable models
• Q. Liu -- Tout groupe fini est un groupe de Galois sur $$\mathbb Q_p(T)$$, d'après Harbater
• W. K. Seiler -- Specializations of coverings and their Galois groups
• L. Wang -- Rational points and canonical heights on K3-surfaces in $$\mathbb P^1\times \mathbb P^1\times \mathbb P^1$$
Part D. Group theory and geometric monodromy groups
• S. S. Abhyankar -- Mathieu group coverings and linear group coverings
• P. Feit -- Fundamental groups for arbitrary categories
• R. M. Guralnick and M. G. Neubauer -- Monodromy groups of branched coverings: The generic case
• D. Harbater -- Fundamental groups and embedding problems in characteristic $$p$$
• M. Jarden -- On free profinite groups of uncountable rank
• P. Müller -- Primitive monodromy groups of polynomials