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BL
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| Record Number
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882504
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| Main Entry
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Malle, Gunter
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| Title & Author
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Inverse Galois theory /\ Gunter Malle, B. Heinrich Matzat.
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| Edition Statement
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Second edition.
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| Publication Statement
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Berlin, Germany :: Springer,, 2018.
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| Series Statement
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Springer monographs in mathematics
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| Page. NO
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1 online resource
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| ISBN
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3662554208
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: 9783662554203
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3662554194
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9783662554197
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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Intro; Preface; Contents; I The Rigidity Method; 1 The Inverse Galois Problem over C(t) and R(t); 1.1 The Fundamental Group of the Punctured RiemannSphere; 1.2 The Algebraic Variant of the Fundamental Group; 1.3 Extension by Complex Conjugation; 1.4 Generalization to Function Fields of Riemann Surfaces; 2 Arithmetic Fundamental Groups; 2.1 Descent to Algebraically Closed Subfields; 2.2 The Fundamental Splitting Sequence; 2.3 The Action via the Cyclotomic Character; 2.4 The Theorem of Belyi; 3 Fields of Definition of Galois Extensions; 3.1 Cyclic and Projective Descent
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1.2 Rigidity for GLn(q)1.3 Galois Realizations for Linear Groups; 2 Pseudo-Reflection Groups and Belyi Triples; 2.1 Groups Generated by Pseudo-Reflections; 2.2 An Effective Version of Belyi's Criterion; 2.3 Imprimitive and Symmetric Groups; 2.4 Invariant Forms; 3 The Classical Groups; 3.1 Rigidity for GUn(q); 3.2 Rigidity for CSp2n(q); 3.3 Rigidity for SO2n+1(q); 3.4 Rigidity for CO2n+(q); 3.5 Rigidity for CO2n-(q); 4 The Exceptional Groups of Rank at Most 2; 4.1 Divisibility Criteria; 4.2 Rigidity for the Ree Groups 2G2(q2); 4.3 Rigidity for the Groups G2(q)
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3.2 Fields of Definition of Geometric Field Extensions3.3 Fields of Definition of Geometric Galois Extensions; 4 The Rigidity Property; 4.1 The Hurwitz Classification; 4.2 The Fixed Field of a Class of Generating Systems; 4.3 The Basic Rigidity Theorem; 4.4 Choice of Ramification Points; 5 Verification of Rigidity; 5.1 Geometric Galois Extensions over Q(t)with Abelian Groups; 5.2 Geometric Galois Extensions over Q(t) with Sn and An; 5.3 Structure Constants; 5.4 The Rigidity Criterion of Belyi; 6 Geometric Automorphisms; 6.1 Extension of the Algebraic Fundamental Group
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6.2 The Action of Geometric Automorphisms6.3 Rigid Orbits; 6.4 The Twisted Rigidity Theorem; 6.5 Geometric Galois Extensions over Q(t) with M12and M11; 7 Rational Translates of Galois Extensions; 7.1 Galois Rational Translates; 7.2 Rational Translates with Few Ramification Points; 7.3 Twisting Rational Translates; 7.4 Geometric Galois Extensions over Q(t) with L2(p); 8 Automorphisms of the Galois Group; 8.1 Fixed Fields of Coarse Classes of Generating Systems; 8.2 Extension of the Galois Group by Outer Automorphisms; 8.3 Geometric Extension of the Galois Group by Outer Automorphisms
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8.4 Geometric Galois Extensions over Q(t) with PGL2(p)9 Computation of Polynomials with Prescribed Group; 9.1 Decomposition of Prime Divisors in Galois Extensions; 9.2 Polynomials with Groups Sn and An; 9.3 Polynomials with the Group Aut(A6) and RelatedGroups; 9.4 Polynomials with the Mathieu Groups M12 and M11; 10 Specialization of Geometric Galois Extensions; 10.1 Local Structure Stability; 10.2 Reality Questions; 10.3 Ramification in Minimal Fields of Definition; 10.4 Ramification in Residue Fields; II Applications of Rigidity; 1 The General Linear Groups; 1.1 Groups of Lie Type
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| Abstract
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This second edition addresses the question of which finite groups occur as Galois groups over a given field. In particular, this includes the question of the structure and the representations of the absolute Galois group of K, as well as its finite epimorphic images, generally referred to as the inverse problem of Galois theory. In the past few years, important strides have been made in all of these areas. The aim of the book is to provide a systematic and extensive overview of these advances, with special emphasis on the rigidity method and its applications. Among others, the book presents the most successful known existence theorems and construction methods for Galois extensions and solutions of embedding problems, together with a collection of the current Galois realizations. There have been two major developments since the first edition of this book was released. The first is the algebraization of the Katz algorithm for (linearly) rigid generating systems of finite groups; the second is the emergence of a modular Galois theory. The latter has led to new construction methods for additive polynomials with given Galois group over fields of positive characteristic. Both methods have their origin in the Galois theory of differential and difference equations.
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| Subject
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Inverse Galois theory.
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| Subject
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Inverse Galois theory.
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| Subject
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MATHEMATICS-- Algebra-- Intermediate.
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| Dewey Classification
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512/.32
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| LC Classification
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QA247
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| Added Entry
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Matzat, B. Heinrich, (Bernd Heinrich),1945-
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