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Título : Algebra : Fields and Galois Theory Tipo de documento: documento electrónico Autores: Falko Lorenz ; SpringerLink (Online service) Editorial: New York, NY : Springer New York Fecha de publicación: 2006 Colección: Universitext, ISSN 0172-5939 Número de páginas: VIII, 296 p. 6 illus Il.: online resource ISBN/ISSN/DL: 978-0-387-31608-6 Idioma : Inglés (eng) Palabras clave: Mathematics Algebra Commutative algebra rings Field theory (Physics) Number Theory and Polynomials Rings Algebras Clasificación: 51 Matemáticas Resumen: The present textbook is a lively, problem-oriented and carefully written introduction to classical modern algebra. The author leads the reader through interesting subject matter, while assuming only the background provided by a first course in linear algebra. The first volume focuses on field extensions. Galois theory and its applications are treated more thoroughly than in most texts. It also covers basic applications to number theory, ring extensions and algebraic geometry. The main focus of the second volume is on additional structure of fields and related topics. Much material not usually covered in textbooks appears here, including real fields and quadratic forms, diophantine dimensions of a field, the calculus of Witt vectors, the Schur group of a field, and local class field theory. Both volumes contain numerous exercises and can be used as a textbook for advanced undergraduate students. From Reviews of the German version: This is a charming textbook, introducing the reader to the classical parts of algebra. The exposition is admirably clear and lucidly written with only minimal prerequisites from linear algebra. The new concepts are, at least in the first part of the book, defined in the framework of the development of carefully selected problems. - Stefan Porubsky, Mathematical Reviews Nota de contenido: Constructibility with Ruler and Compass -- Algebraic Extensions -- Simple Extensions -- Fundamentals of Divisibility -- Prime Factorization in Polynomial Rings. Gauss’s Theorem -- Polynomial Splitting Fields -- Separable Extensions -- Galois Extensions -- Finite Fields, Cyclic Groups and Roots of Unity -- Group Actions -- Applications of Galois Theory to Cyclotomic Fields -- Further Steps into Galois Theory -- Norm and Trace -- Binomial Equations -- Solvability of Equations -- Integral Ring Extensions with Applications to Galois Theory -- The Transcendence of ? -- Fundamentals of Transcendental Field Extensions -- Hilbert’s Nullstellensatz En línea: http://dx.doi.org/10.1007/0-387-31608-6 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=34789 Algebra : Fields and Galois Theory [documento electrónico] / Falko Lorenz ; SpringerLink (Online service) . - New York, NY : Springer New York, 2006 . - VIII, 296 p. 6 illus : online resource. - (Universitext, ISSN 0172-5939) .
ISBN : 978-0-387-31608-6
Idioma : Inglés (eng)
Palabras clave: Mathematics Algebra Commutative algebra rings Field theory (Physics) Number Theory and Polynomials Rings Algebras Clasificación: 51 Matemáticas Resumen: The present textbook is a lively, problem-oriented and carefully written introduction to classical modern algebra. The author leads the reader through interesting subject matter, while assuming only the background provided by a first course in linear algebra. The first volume focuses on field extensions. Galois theory and its applications are treated more thoroughly than in most texts. It also covers basic applications to number theory, ring extensions and algebraic geometry. The main focus of the second volume is on additional structure of fields and related topics. Much material not usually covered in textbooks appears here, including real fields and quadratic forms, diophantine dimensions of a field, the calculus of Witt vectors, the Schur group of a field, and local class field theory. Both volumes contain numerous exercises and can be used as a textbook for advanced undergraduate students. From Reviews of the German version: This is a charming textbook, introducing the reader to the classical parts of algebra. The exposition is admirably clear and lucidly written with only minimal prerequisites from linear algebra. The new concepts are, at least in the first part of the book, defined in the framework of the development of carefully selected problems. - Stefan Porubsky, Mathematical Reviews Nota de contenido: Constructibility with Ruler and Compass -- Algebraic Extensions -- Simple Extensions -- Fundamentals of Divisibility -- Prime Factorization in Polynomial Rings. Gauss’s Theorem -- Polynomial Splitting Fields -- Separable Extensions -- Galois Extensions -- Finite Fields, Cyclic Groups and Roots of Unity -- Group Actions -- Applications of Galois Theory to Cyclotomic Fields -- Further Steps into Galois Theory -- Norm and Trace -- Binomial Equations -- Solvability of Equations -- Integral Ring Extensions with Applications to Galois Theory -- The Transcendence of ? -- Fundamentals of Transcendental Field Extensions -- Hilbert’s Nullstellensatz En línea: http://dx.doi.org/10.1007/0-387-31608-6 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=34789 Ejemplares
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Título : Algebraic Theory of Quadratic Numbers Tipo de documento: documento electrónico Autores: Mak Trifkovic ; SpringerLink (Online service) Editorial: New York, NY : Springer New York Fecha de publicación: 2013 Otro editor: Imprint: Springer Colección: Universitext, ISSN 0172-5939 Número de páginas: XI, 197 p. 29 illus Il.: online resource ISBN/ISSN/DL: 978-1-4614-7717-4 Idioma : Inglés (eng) Palabras clave: Mathematics Algebra Number theory Theory Clasificación: 51 Matemáticas Resumen: By focusing on quadratic numbers, this advanced undergraduate or master’s level textbook on algebraic number theory is accessible even to students who have yet to learn Galois theory. The techniques of elementary arithmetic, ring theory and linear algebra are shown working together to prove important theorems, such as the unique factorization of ideals and the finiteness of the ideal class group. The book concludes with two topics particular to quadratic fields: continued fractions and quadratic forms. The treatment of quadratic forms is somewhat more advanced than usual, with an emphasis on their connection with ideal classes and a discussion of Bhargava cubes. The numerous exercises in the text offer the reader hands-on computational experience with elements and ideals in quadratic number fields. The reader is also asked to fill in the details of proofs and develop extra topics, like the theory of orders. Prerequisites include elementary number theory and a basic familiarity with ring theory Nota de contenido: 1 Examples -- 2 A Crash Course in Ring Theory -- 3 Lattices -- 4 Arithmetic in Q[vD] -- 5 The Ideal Class Group and Geometry of Numbers -- 6 Continued Fractions -- 7 Quadratic Forms -- Appendix -- Hints to Selected Exercises -- Index En línea: http://dx.doi.org/10.1007/978-1-4614-7717-4 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=32363 Algebraic Theory of Quadratic Numbers [documento electrónico] / Mak Trifkovic ; SpringerLink (Online service) . - New York, NY : Springer New York : Imprint: Springer, 2013 . - XI, 197 p. 29 illus : online resource. - (Universitext, ISSN 0172-5939) .
ISBN : 978-1-4614-7717-4
Idioma : Inglés (eng)
Palabras clave: Mathematics Algebra Number theory Theory Clasificación: 51 Matemáticas Resumen: By focusing on quadratic numbers, this advanced undergraduate or master’s level textbook on algebraic number theory is accessible even to students who have yet to learn Galois theory. The techniques of elementary arithmetic, ring theory and linear algebra are shown working together to prove important theorems, such as the unique factorization of ideals and the finiteness of the ideal class group. The book concludes with two topics particular to quadratic fields: continued fractions and quadratic forms. The treatment of quadratic forms is somewhat more advanced than usual, with an emphasis on their connection with ideal classes and a discussion of Bhargava cubes. The numerous exercises in the text offer the reader hands-on computational experience with elements and ideals in quadratic number fields. The reader is also asked to fill in the details of proofs and develop extra topics, like the theory of orders. Prerequisites include elementary number theory and a basic familiarity with ring theory Nota de contenido: 1 Examples -- 2 A Crash Course in Ring Theory -- 3 Lattices -- 4 Arithmetic in Q[vD] -- 5 The Ideal Class Group and Geometry of Numbers -- 6 Continued Fractions -- 7 Quadratic Forms -- Appendix -- Hints to Selected Exercises -- Index En línea: http://dx.doi.org/10.1007/978-1-4614-7717-4 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=32363 Ejemplares
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Título : An Introduction to Homological Algebra Tipo de documento: documento electrónico Autores: Joseph J. Rotman ; SpringerLink (Online service) Editorial: New York, NY : Springer New York Fecha de publicación: 2009 Colección: Universitext, ISSN 0172-5939 Número de páginas: XIV, 710 p. 11 illus Il.: online resource ISBN/ISSN/DL: 978-0-387-68324-9 Idioma : Inglés (eng) Palabras clave: Mathematics Category theory (Mathematics) Homological algebra Theory, Algebra Clasificación: 51 Matemáticas Resumen: With a wealth of examples as well as abundant applications to Algebra, this is a must-read work: a clearly written, easy-to-follow guide to Homological Algebra. The author provides a treatment of Homological Algebra which approaches the subject in terms of its origins in algebraic topology. In this brand new edition the text has been fully updated and revised throughout and new material on sheaves and abelian categories has been added. Applications include the following: * to rings -- Lazard's theorem that flat modules are direct limits of free modules, Hilbert's Syzygy Theorem, Quillen-Suslin's solution of Serre's problem about projectives over polynomial rings, Serre-Auslander-Buchsbaum characterization of regular local rings (and a sketch of unique factorization); * to groups -- Schur-Zassenhaus, Gaschutz's theorem on outer automorphisms of finite p-groups, Schur multiplier, cotorsion groups; * to sheaves -- sheaf cohomology, Cech cohomology, discussion of Riemann-Roch Theorem over compact Riemann surfaces. Learning Homological Algebra is a two-stage affair. Firstly, one must learn the language of Ext and Tor, and what this describes. Secondly, one must be able to compute these things using a separate language: that of spectral sequences. The basic properties of spectral sequences are developed using exact couples. All is done in the context of bicomplexes, for almost all applications of spectral sequences involve indices. Applications include Grothendieck spectral sequences, change of rings, Lyndon-Hochschild-Serre sequence, and theorems of Leray and Cartan computing sheaf cohomology. Joseph Rotman is Professor Emeritus of Mathematics at the University of Illinois at Urbana-Champaign. He is the author of numerous successful textbooks, including Advanced Modern Algebra (Prentice-Hall 2002), Galois Theory, 2nd Edition (Springer 1998) A First Course in Abstract Algebra (Prentice-Hall 1996), Introduction to the Theory of Groups, 4th Edition (Springer 1995), and Introduction to Algebraic Topology (Springer 1988) Nota de contenido: Hom and Tensor -- Special Modules -- Specific Rings -- Setting the Stage -- Homology -- Tor and Ext -- Homology and Rings -- Homology and Groups -- Spectral Sequences En línea: http://dx.doi.org/10.1007/b98977 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=33837 An Introduction to Homological Algebra [documento electrónico] / Joseph J. Rotman ; SpringerLink (Online service) . - New York, NY : Springer New York, 2009 . - XIV, 710 p. 11 illus : online resource. - (Universitext, ISSN 0172-5939) .
ISBN : 978-0-387-68324-9
Idioma : Inglés (eng)
Palabras clave: Mathematics Category theory (Mathematics) Homological algebra Theory, Algebra Clasificación: 51 Matemáticas Resumen: With a wealth of examples as well as abundant applications to Algebra, this is a must-read work: a clearly written, easy-to-follow guide to Homological Algebra. The author provides a treatment of Homological Algebra which approaches the subject in terms of its origins in algebraic topology. In this brand new edition the text has been fully updated and revised throughout and new material on sheaves and abelian categories has been added. Applications include the following: * to rings -- Lazard's theorem that flat modules are direct limits of free modules, Hilbert's Syzygy Theorem, Quillen-Suslin's solution of Serre's problem about projectives over polynomial rings, Serre-Auslander-Buchsbaum characterization of regular local rings (and a sketch of unique factorization); * to groups -- Schur-Zassenhaus, Gaschutz's theorem on outer automorphisms of finite p-groups, Schur multiplier, cotorsion groups; * to sheaves -- sheaf cohomology, Cech cohomology, discussion of Riemann-Roch Theorem over compact Riemann surfaces. Learning Homological Algebra is a two-stage affair. Firstly, one must learn the language of Ext and Tor, and what this describes. Secondly, one must be able to compute these things using a separate language: that of spectral sequences. The basic properties of spectral sequences are developed using exact couples. All is done in the context of bicomplexes, for almost all applications of spectral sequences involve indices. Applications include Grothendieck spectral sequences, change of rings, Lyndon-Hochschild-Serre sequence, and theorems of Leray and Cartan computing sheaf cohomology. Joseph Rotman is Professor Emeritus of Mathematics at the University of Illinois at Urbana-Champaign. He is the author of numerous successful textbooks, including Advanced Modern Algebra (Prentice-Hall 2002), Galois Theory, 2nd Edition (Springer 1998) A First Course in Abstract Algebra (Prentice-Hall 1996), Introduction to the Theory of Groups, 4th Edition (Springer 1995), and Introduction to Algebraic Topology (Springer 1988) Nota de contenido: Hom and Tensor -- Special Modules -- Specific Rings -- Setting the Stage -- Homology -- Tor and Ext -- Homology and Rings -- Homology and Groups -- Spectral Sequences En línea: http://dx.doi.org/10.1007/b98977 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=33837 Ejemplares
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Título : An Introduction to Manifolds Tipo de documento: documento electrónico Autores: Loring W. Tu ; SpringerLink (Online service) Editorial: New York, NY : Springer New York Fecha de publicación: 2008 Colección: Universitext, ISSN 0172-5939 Número de páginas: XVI, 368 p. 104 illus Il.: online resource ISBN/ISSN/DL: 978-0-387-48101-2 Idioma : Inglés (eng) Palabras clave: Mathematics Global analysis (Mathematics) Manifolds Differential geometry Complex manifolds and Cell Complexes (incl. Diff.Topology) Analysis on Geometry Clasificación: 51 Matemáticas Resumen: Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, An Introduction to Manifolds is also an excellent foundation for Springer GTM 82, Differential Forms in Algebraic Topology. Nota de contenido: Euclidean Spaces -- Smooth Functions on a Euclidean Space -- Tangent Vectors in Rn as Derivations -- Alternating k-Linear Functions -- Differential Forms on Rn -- Manifolds -- Manifolds -- Smooth Maps on a Manifold -- Quotients -- Lie Groups and Lie Algebras -- The Tangent Space -- Submanifolds -- Categories and Functors -- The Rank of a Smooth Map -- The Tangent Bundle -- Bump Functions and Partitions of Unity -- Vector Fields -- Lie Groups and Lie Algebras -- Lie Groups -- Lie Algebras -- Differential Forms -- Differential 1-Forms -- Differential k-Forms -- The Exterior Derivative -- Integration -- Orientations -- Manifolds with Boundary -- Integration on a Manifold -- De Rham Theory -- De Rham Cohomology -- The Long Exact Sequence in Cohomology -- The Mayer–Vietoris Sequence -- Homotopy Invariance -- Computation of de Rham Cohomology -- Proof of Homotopy Invariance En línea: http://dx.doi.org/10.1007/978-0-387-48101-2 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=34148 An Introduction to Manifolds [documento electrónico] / Loring W. Tu ; SpringerLink (Online service) . - New York, NY : Springer New York, 2008 . - XVI, 368 p. 104 illus : online resource. - (Universitext, ISSN 0172-5939) .
ISBN : 978-0-387-48101-2
Idioma : Inglés (eng)
Palabras clave: Mathematics Global analysis (Mathematics) Manifolds Differential geometry Complex manifolds and Cell Complexes (incl. Diff.Topology) Analysis on Geometry Clasificación: 51 Matemáticas Resumen: Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, An Introduction to Manifolds is also an excellent foundation for Springer GTM 82, Differential Forms in Algebraic Topology. Nota de contenido: Euclidean Spaces -- Smooth Functions on a Euclidean Space -- Tangent Vectors in Rn as Derivations -- Alternating k-Linear Functions -- Differential Forms on Rn -- Manifolds -- Manifolds -- Smooth Maps on a Manifold -- Quotients -- Lie Groups and Lie Algebras -- The Tangent Space -- Submanifolds -- Categories and Functors -- The Rank of a Smooth Map -- The Tangent Bundle -- Bump Functions and Partitions of Unity -- Vector Fields -- Lie Groups and Lie Algebras -- Lie Groups -- Lie Algebras -- Differential Forms -- Differential 1-Forms -- Differential k-Forms -- The Exterior Derivative -- Integration -- Orientations -- Manifolds with Boundary -- Integration on a Manifold -- De Rham Theory -- De Rham Cohomology -- The Long Exact Sequence in Cohomology -- The Mayer–Vietoris Sequence -- Homotopy Invariance -- Computation of de Rham Cohomology -- Proof of Homotopy Invariance En línea: http://dx.doi.org/10.1007/978-0-387-48101-2 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=34148 Ejemplares
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Título : An Introduction to Manifolds Tipo de documento: documento electrónico Autores: Loring W. Tu ; SpringerLink (Online service) Editorial: New York, NY : Springer New York Fecha de publicación: 2011 Colección: Universitext, ISSN 0172-5939 Número de páginas: XVIII, 410 p. 124 illus., 1 illus. in color Il.: online resource ISBN/ISSN/DL: 978-1-4419-7400-6 Idioma : Inglés (eng) Palabras clave: Mathematics Global analysis (Mathematics) Manifolds Differential geometry Complex manifolds and Cell Complexes (incl. Diff.Topology) Analysis on Geometry Clasificación: 51 Matemáticas Resumen: Manifolds, the higher-dimensional analogues of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way the reader acquires the knowledge and skills necessary for further study of geometry and topology. The second edition contains fifty pages of new material. Many passages have been rewritten, proofs simplified, and new examples and exercises added. This work may be used as a textbook for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. The requisite point-set topology is included in an appendix of twenty-five pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. Requiring only minimal undergraduate prerequisites, "An Introduction to Manifolds" is also an excellent foundation for the author's publication with Raoul Bott, "Differential Forms in Algebraic Topology." Nota de contenido: Preface to the Second Edition -- Preface to the First Edition -- Chapter 1. Euclidean Spaces -- Chapter 2. Manifolds -- Chapter 3. The Tangent Space -- Chapter 4. Lie Groups and Lie Algebras.-Chapter 5. Differential Forms -- Chapter 6. Integration.-Chapter 7. De Rham Theory -- Appendices -- A. Point-Set Topology -- B. The Inverse Function Theorem on R(N) and Related Results -- C. Existence of a Partition of Unity in General -- D. Linear Algebra -- E. Quaternions and the Symplectic Group -- Solutions to Selected Exercises -- Hints and Solutions to Selected End-of-Section Problems -- List of Symbols -- References -- Index En línea: http://dx.doi.org/10.1007/978-1-4419-7400-6 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=33161 An Introduction to Manifolds [documento electrónico] / Loring W. Tu ; SpringerLink (Online service) . - New York, NY : Springer New York, 2011 . - XVIII, 410 p. 124 illus., 1 illus. in color : online resource. - (Universitext, ISSN 0172-5939) .
ISBN : 978-1-4419-7400-6
Idioma : Inglés (eng)
Palabras clave: Mathematics Global analysis (Mathematics) Manifolds Differential geometry Complex manifolds and Cell Complexes (incl. Diff.Topology) Analysis on Geometry Clasificación: 51 Matemáticas Resumen: Manifolds, the higher-dimensional analogues of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way the reader acquires the knowledge and skills necessary for further study of geometry and topology. The second edition contains fifty pages of new material. Many passages have been rewritten, proofs simplified, and new examples and exercises added. This work may be used as a textbook for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. The requisite point-set topology is included in an appendix of twenty-five pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. Requiring only minimal undergraduate prerequisites, "An Introduction to Manifolds" is also an excellent foundation for the author's publication with Raoul Bott, "Differential Forms in Algebraic Topology." Nota de contenido: Preface to the Second Edition -- Preface to the First Edition -- Chapter 1. Euclidean Spaces -- Chapter 2. Manifolds -- Chapter 3. The Tangent Space -- Chapter 4. Lie Groups and Lie Algebras.-Chapter 5. Differential Forms -- Chapter 6. Integration.-Chapter 7. De Rham Theory -- Appendices -- A. Point-Set Topology -- B. The Inverse Function Theorem on R(N) and Related Results -- C. Existence of a Partition of Unity in General -- D. Linear Algebra -- E. Quaternions and the Symplectic Group -- Solutions to Selected Exercises -- Hints and Solutions to Selected End-of-Section Problems -- List of Symbols -- References -- Index En línea: http://dx.doi.org/10.1007/978-1-4419-7400-6 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=33161 Ejemplares
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