1996/08/13 by Helmut Völklein · 2 citations
Mathematics · #History and Theory of Mathematics #Algebraic Geometry and Number Theory #Galois theory #Mathematics #Fundamental theorem of Galois theory #Galois group #Braid group #Group (periodic table) #Algebra over a field #Galois extension #Riemann surface #Group theory #Galois module #Embedding problem #Pure mathematics #Group action #Abstract algebra #Riemann–Hurwitz formula #Geometric function theory
paper · doi:10.1017/cbo9780511471117
openalex publication_date 1996/08/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/06/30
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 realisation 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 mathematicians from other areas (especially group theory) will find this an excellent introduction to a fascinating field.