Algebra Books

Book SynopsisFrom the contents:Preface for the English edition; V.I. Arnold.- Preface.- Introduction.- 1: Groups.- 2: The complex numbers.- 3: Hints, Solutions and Answers.- Appendix. Solvability of equations by explicit formulae; A. Khovanskii.- Bibliography.- Appendix; V.I. Arnold.- Index. Trade ReviewFrom the reviews: "This very special and brilliant text has been written for bright non-specialists in mathematics, but it leads the reader up to topical research problems in the field, and that in a masterly manner. The book is absolutely self-contained, in its own particular fashion, and it is therefore perfectly suited for self-study, ranging from advanced high school to graduate level. No doubt, the thorough and serious working with this outstanding text could turn very beginners into creative almost-experts in the field." (Werner Kleinert, Zentralblatt MATH, Vol. 1065 (16), 2005)Table of ContentsPreface for the English edition; V.I. Arnold. Preface. Introduction. 1: Groups. 1.1. Examples. 1.2. Groups of transformations. 1.3. Groups. 1.4. Cyclic groups. 1.5. Isomorphisms. 1.6. Subgroups. 1.7. Direct product. 1.8. Cosets. Lagrange's theory. 1.9. Internal automorphisms. 1.10. Normal subgroups. 1.11. Quotient groups. 1.12. Commutant. 1.13. Homomorphisms. 1.14. Soluble groups. 1.15. Permutations. 2: The complex numbers. 2.1. Fields and polynomials. 2.2. The field of complex numbers. 2.3. Uniqueness of the field of complex numbers. 2.4. Geometrical descriptions of the field of complex numbers. 2.5. The trigonometric form of the complex numbers. 2.6. Continuity. 2.7. Continuous curves. 2.8. Images of curves: the basic theorem of the algebra of complex numbers. 2.9. The Riemann surface of the function w = SQRTz. 2.10. The Riemann surfaces of more complicated functions. 2.11. Functions representable by radicals. 2.12. Monodromy groups of multi-valued functions. 2.13. Monodromy groups of functions representable by radicals. 2.14. The Abel theorem. 3: Hints, Solutions and Answers. 3.1.Problems of Chapter 1. 3.2. Problems of Chapter 2. Drawings of Riemann surfaces; F. Aicardi. Appendix. Solvability of equations by explicit formulae; A. Khovanskii. A.1. Explicit solvability of equations. A.2. Liouville's theory. A.3. Picard-Vessiot's theory. A.4. Topological obstructions for the representation of functions by quadratures. A.5. S-functions. A.6. Monodromy group. A.7. Obstructions for the representability of functions by quadratures. A.8. Solvability of algebraic equations. A.9. The monodromy pair. A.10. Mapping of the semi-plane to a polygon bounded by arcs of circles. A.11. Topological obstructions for the solvability of differential equations. A.12. Algebraic functions of several variables. A.13. Functions of several complex variables representable by quadratures and generalized quadratures. A.14. SC-germs. A.15. Topological obstruction for the solvability of the holonomic systems of linear differential equations. A.16. Topological obstruction for the solvability of the holonomic systems of linear differential equations. Bibliography. Appendix; V.I. Arnold. Index.

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Book SynopsisStraightedge and compass.- Euclid's approach to geometry.- Coordinates.- Vectors and Euclidean spaces.- Perspective.- Projective planes.- Transformations.- Non-Euclidean geometry.Trade ReviewFrom the reviews:"This is an introductory book on geometry, easy to read, written in an engaging style. The author’s goal is … to increase one’s overall understanding and appreciation of the subject. … Along the way, he presents elegant proofs of well-known theorems … . The advantage of the author’s approach is clear: in a short space he gives a brief introduction to many sides of geometry and includes many beautiful results, each explained from a perspective that makes it easy to understand." (Robin Hartshorne, SIAM Review, Vol. 48 (2), 2006)"The pillars of the title are … Euclidean construction and axioms, coordinates and vectors, projective geometry, and transformations and non-Euclidean geometry. … The writing style is both student-friendly and deeply informed. The pleasing brevity of the book … makes the book especially suitable as an instruction to geometry for the large and critically important population of undergraduate mathematics majors … . Each chapter concludes with a well-written discussion section that combines history with glances at further results. There is a good selection of thought-provoking exercises." (R. J. Bumcrot, Mathematical Reviews, Issue 2006 e)"The author acts on the assumption of four approaches to geometry: The axiomatic way, using linear Algebra, projective geometry and transformation groups. … Each of the chapters closes with a discussion giving hints on further aspects and historical remarks. … The book can be recommended to be used in undergraduate courses on geometry … ." (F. Manhart, Internationale Mathematische Nachrichten, Issue 203, 2006)"Any new mathematics textbook by John Stillwell is worth a serious look. Stillwell is the prolific author of more than half a dozen textbooks … . I would not hesitate to recommend this text to any professor teaching a course in geometry who is more interested in providing a rapid survey of topics rather than an in-depth, semester-long, examination of any particular one." (Mark Hunacek, The Mathematical Gazette, Vol. 91 (521), 2007)"The title refers to four different approaches to elementary geometry which according to the author only together show this field in all its splendor: via straightedge and compass constructions, linear algebra, projective geometry and transformation groups. … the book can be recommended warmly to undergraduates to get in touch with geometric thinking." (G. Kowol, Monatshefte für Mathematik, Vol. 150 (3), 2007)"This book presents a tour on various approaches to a notion of geometry and the relationship between these approaches. … The book shows clearly how useful it is to use various tools in a description of basic geometrical questions to find the simplest and the most intuitive arguments for different problems. The book is a very useful source of ideas for high school teachers." (EMS Newsletter, March, 2007)“The four pillars of geometry approaches geometry in four different ways, devoting two chapters to each, the first chapter being concrete and introductory, the second more abstract. … The content is quite elementary and is based on lectures given by the author at the University of San Francisco in 2004. … The book of Stillwell is a very good first introduction to geometry especially for the axiomatic and the projective point of view.” (Yves Félix, Bulletin of the Belgian Mathematical Society, Vol. 15 (1), 2008)Table of ContentsStraightedge and compass.- Euclid’s approach to geometry.- Coordinates.- Vectors and Euclidean spaces.- Perspective.- Projective planes.- Transformations.- Non-Euclidean geometry.

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Book Synopsis1. Sets.- 2. Groups.- 3. Lattices and Categories.- 4. Rings and Modules.- 5. Algebras.- 6. Multilinear Algebra.- 7. Field Theory.- 8. Quadratic Forms and Ordered Fields.- 9. Valuation Theory.- 10. Commutative Rings.- 11. Infinite Field Extensions.- List of Notations.- Author Index.Trade ReviewFrom the reviews: "The book is a wonderful piece of work that condenses and transmits the long experience of an excellent teacher. The reader enjoys the structure of the precisely designed volume and the beautiful combination of clear arguments and well-chosen examples. A lot of exercises help the reader to deepen his knowledge. This updated and improved introduction to abstract algebra must be on the bookshelves of all algebraists and of all students interested in algebra." (Maria B. Szendrei, Acta Scientiarum Mathematicarum, Vol. 71, 2005) "This is a coherent overview of group, ring and field theory which combines brevity with elegance and authority. … to the serious algebraists and departmental libraries … this book will be an automatic purchase." (Gerry Leversha, The Mathematical Gazette, March, 2005) "This book … is a new and revised version of the author’s famous text ‘Algebra’ … . The book contains numerous exercises, and … many illuminating comments on the subject. There is no doubt that the book will take the position of its predecessor in being one of the most outstanding introductory algebra textbooks." (EMS Newsletter, June, 2004) "The entire text is profusely supported by worked examples. Each section in the book comes with a plentiful supply of carefully selected exercises, and there are also numerous historical remarks and hints for further reading. Altogether, this textbook breathes once more the author’s rich teaching experience and his masterly skill as a textbook writer. … the author has kept his well-tried methodical principle of combining old and new viewpoints in algebra in a natural way, which makes his textbooks so unique, matchless and timelessly valuable." (Werner Kleinert, Zentralblatt MATH, Vol. 1003 (3), 2003)Table of ContentsPreface.- Conventions on Terminology.- Sets.- Groups.- Lattices and Categories.- Rings and Modules.- Algebras.- Multilinear Algebra.- Field Theory.- Quadratic Forms and Ordered Fields.- Valuation Theory.- Commutative Rings.- Infinite Field Extensions.- Bibliography.- List of Notations.- Author Index.- Subject Index.

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Book Synopsis1. Universal algebra.- 2. Homological algebra.- 3. Further group theory.- 4. Algebras.- 5. Central simple algebras.- 6. Representation theory of finite groups.- 7. Noetherian rings and polynomial identities.- 8. Rings without finiteness assumptions.- 9. Skew fields.- 10. Coding theory.- 11. Languages and automata.- List of Notations.- Author Index.Trade ReviewFrom the reviews of the second volume of the revised edition: "The reader is treated to the spectacle of a very fine mathematician developing many diverse branches of algebra in a direct and illuminating manner. … this volume contains a gold-mine of algebraic nuggets, each developed in a direct and economical manner highlighting the most important results. It is an invaluable source and is to be strongly recommended." (Robert Curtis, The Mathematical Gazette, Vol. 88 (512), 2004) "The author has spent much effort at including a plentiful supply of worked concrete examples and carefully selected exercises. … it is especially the wealth of specific, non-standard topics and important applications that makes this algebra text … highly unique and valuable. … the author manages to cover a vast spectrum of concepts, methods, principles, aspects, and applications of modern algebra in a masterly style. … This introductory algebra text remains one of the very best available, all the more … in this new edition." (Werner Kleinert, Zentralblatt MATH, 1006, 2003)Table of Contents1. Universal algebra.- 2. Homological algebra.- 3. Further group theory.- 4. Algebras.- 5. Central simple algebras.- 6. Representation theory of finite groups.- 7. Noetherian rings and polynomial identities.- 8. Rings without finiteness assumptions.- 9. Skew fields.- 10. Coding theory.- 11. Languages and automata.- List of Notations.- Author Index.

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Book SynopsisI. Spaces.- 1. Fields.- 2. Vector spaces.- 3. Examples.- 4. Comments.- 5. Linear dependence.- 6. Linear combinations.- 7. Bases.- 8. Dimension.- 9. Isomorphism.- 10. Subspaces.- 11. Calculus of subspaces.- 12. Dimension of a subspace.- 13. Dual spaces.- 14. Brackets.- 15. Dual bases.- 16. Reflexivity.- 17. Annihilators.- 18. Direct sums.- 19. Dimension of a direct sum.- 20. Dual of a direct sum.- 21. Quotient spaces.- 22. Dimension of a quotient space.- 23. Bilinear forms.- 24. Tensor products.- 25. Product bases.- 26. Permutations.- 27. Cycles.- 28. Parity.- 29. Multilinear forms.- 30. Alternating forms.- 31. Alternating forms of maximal degree.- II. Transformations.- 32. Linear transformations.- 33. Transformations as vectors.- 34. Products.- 35. Polynomials.- 36. Inverses.- 37. Matrices.- 38. Matrices of transformations.- 39. Invariance.- 40. Reducibility.- 41. Projections.- 42. Combinations of projections.- 43. Projections and invariance.- 44. Adjoints.- 45. Adjoints of projectionsTrade Review“This is a classic but still useful introduction to modern linear algebra. It is primarily about linear transformations … . It’s also extremely well-written and logical, with short and elegant proofs. … The exercises are very good, and are a mixture of proof questions and concrete examples. The book ends with a few applications to analysis … and a brief summary of what is needed to extend this theory to Hilbert spaces.” (Allen Stenger, MAA Reviews, maa.org, May, 2016)“The theory is systematically developed by the axiomatic method that has, since von Neumann, dominated the general approach to linear functional analysis and that achieves here a high degree of lucidity and clarity. The presentation is never awkward or dry, as it sometimes is in other “modern” textbooks; it is as unconventional as one has come to expect from the author. The book contains about 350 well placed and instructive problems, which cover a considerable part of the subject. All in all this is an excellent work, of equally high value for both student and teacher.” Zentralblatt für MathematikTable of ContentsI. Spaces.- 1. Fields.- 2. Vector spaces.- 3. Examples.- 4. Comments.- 5. Linear dependence.- 6. Linear combinations.- 7. Bases.- 8. Dimension.- 9. Isomorphism.- 10. Subspaces.- 11. Calculus of subspaces.- 12. Dimension of a subspace.- 13. Dual spaces.- 14. Brackets.- 15. Dual bases.- 16. Reflexivity.- 17. Annihilators.- 18. Direct sums.- 19. Dimension of a direct sum.- 20. Dual of a direct sum.- 21. Quotient spaces.- 22. Dimension of a quotient space.- 23. Bilinear forms.- 24. Tensor products.- 25. Product bases.- 26. Permutations.- 27. Cycles.- 28. Parity.- 29. Multilinear forms.- 30. Alternating forms.- 31. Alternating forms of maximal degree.- II. Transformations.- 32. Linear transformations.- 33. Transformations as vectors.- 34. Products.- 35. Polynomials.- 36. Inverses.- 37. Matrices.- 38. Matrices of transformations.- 39. Invariance.- 40. Reducibility.- 41. Projections.- 42. Combinations of pro¬jections.- 43. Projections and invariance.- 44. Adjoints.- 45. Adjoints of projections.- 46. Change of basis.- 47. Similarity.- 48. Quotient transformations.- 49. Range and null-space.- 50. Rank and nullity.- 51. Transformations of rank one.- 52. Tensor products of transformations.- 53. Determinants.- 54. Proper values.- 55. Multiplicity.- 56. Triangular form.- 57. Nilpotence.- 58. Jordan form.- III. Orthogonality.- 59. Inner products.- 60. Complex inner products.- 61. Inner product spaces.- 62. Orthogonality.- 63. Completeness.- 64. Schwarz’s inequality.- 65. Complete orthonormal sets.- 66. Projection theorem.- 67. Linear functionals.- 68. Parentheses versus brackets.- 69. Natural isomorphisms.- 70. Self-adjoint transformations.- 71. Polarization.- 72. Positive transformations.- 73. Isometries.- 74. Change of orthonormal basis.- 75. Perpendicular projections.- 76. Combinations of perpendicular projections.- 77. Complexification.- 78. Characterization of spectra.- 79. Spectral theorem.- 80. Normal transformations.- 81. Orthogonal transformations.- 82. Functions of transformations.- 83. Polar decomposition.- 84. Commutativity.- 85. Self-adjoint transformations of rank one.- IV. Analysis.- 86. Convergence of vectors.- 87. Norm.- 88. Expressions for the norm.- 89. Bounds of a self-adjoint transformation.- 90. Minimax principle.- 91. Convergence of linear transformations.- 92. Ergodic theorem.- 93. Power series.- Appendix. Hilbert Space.- Recommended Reading.- Index of Terms.- Index of Symbols.

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Book SynopsisA look at solving problems in three areas of classical elementary mathematics: equations and systems of equations of various kinds, algebraic inequalities, and elementary number theory, in particular divisibility and diophantine equations. In each topic, brief theoretical discussions are followed by carefully worked out examples of increasing difficulty, and by exercises which range from routine to rather more challenging problems. While it emphasizes some methods that are not usually covered in beginning university courses, the book nevertheless teaches techniques and skills which are useful beyond the specific topics covered here. With approximately 330 examples and 760 exercises.Trade Review“In each topic, brief theoretical discussions are followed by carefully worked out examples of increasing difficulty, and by exercises which range from routine to rather more challenging problems. While it emphasizes some methods that are not usually covered in beginning university courses, the book nevertheless teaches techniques and skills which are useful beyond the specific topics covered here.” (Olaf Ninnemann, zbMATH 1055.00006, 2021)From the reviews: MATHEMATICAL INTELLIGENCER "I have nothing but praise for this book, and I can't imagine a mathamtician who wouldn't want to own it."MAA ONLINE"This book contains over 300 examples and then over 700 exercises for the reader. The authors have even included a fourth chapter with hints and answers to the exercises. So, even if the solution isn't given in the back, at least the reader will have some hint about which direction to start looking. The book provides a nice introduction into many of the problems-solvers' 'tricks of the trade.' There is a lot presented here, so the reader (especially an undergraduate) may want to take the following approach: read just a few pages at a time, work out some of the exercises, then take some time to digest the material before going on." Table of Contents1 Algebraic Identities and Equations.- 1 Formulas for Powers.- 2 Finite Sums.- 3 Polynomials.- 4 Symmetric Polynomials.- 5 Systems of Equations.- 6 Irrational Equations.- 7 Some Applications of Complex Numbers.- 2 Algebraic Inequalities.- 1 Definitions and Properties.- 2 Basic Methods.- 3 The Use of Algebraic Formulas.- 4 The Method of Squares.- 5 The Discriminant and Cauchy’s Inequality.- 6 The Induction Principle.- 7 Chebyshev’s Inequality.- 8 Inequalities Between Means.- 9 Appendix on Irrational Numbers.- 3 Number Theory.- 1 Basic Concepts.- 2 Prime Numbers.- 3 Congruences.- 4 Congruences in One Variable.- 5 Diophantine Equations.- 6 Solvability of Diopha,ntine Equations.- 7 Integer Part and Fractional Part.- 8 Base Representations.- 9 Dirichlet’s Principle.- 10 Polynomials.- 4 Hints and Answers.- 1 Hints and Answers to Chapter 1.- 2 Hints and Answers to Chapter 2.- 3 Hints and Answers to Chapter 3.

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Book SynopsisThe study of associative algebras con­ tributes to and draws from such topics as group theory, commutative ring theory, field theory, algebraic number theory, algebraic geometry, homo­ logical algebra, and category theory.Table of Contents1 The Associative Algebra.- 1.1. Conventions.- 1.2. Group Algebras.- 1.3. Endomorphism Algebras.- 1.4. Matrix Algebras.- 1.5. Finite Dimensional Algebras over a Field.- 1.6. Quaternion Algebras.- 1.7. Isomorphism of Quaternion Algebras.- 2 Modules.- 2.1. Change of Scalars.- 2.2. The Lattice of Submodules.- 2.3. Simple Modules.- 2.4. Semisimple Modules.- 2.5. Structure of Semisimple Modules.- 2.6. Chain Conditions.- 2.7. The Radical.- 3 The Structure of Semisimple Algebras.- 3.1. Semisimple Algebras.- 3.2. Minimal Right Ideals.- 3.3. Simple Algebras.- 3.4. Matrices of Homomorphisms.- 3.5. Wedderbum’s Structure Theorem.- 3.6. Maschke’s Theorem.- 4 The Radical.- 4.1. The Radical of an Algebra.- 4.2. Nakayama’s Lemma.- 4.3. The Jacobson Radical.- 4.4. The Radical of an Artinian Algebra.- 4.5. Artinian Algebras Are Noetherian.- 4.6. Nilpotent Algebras.- 4.7. The Radical of a Group Algebra.- 4.8. Ideals in Artinian Algebras.- 5 Indecomposable Modules.- 5.1. Direct Decompositions.- 5.2. Local Algebras.- 5.3. Fitting’s Lemma.- 5.4. The Krull-Schmidt Theorem.- 5.5. Representations of Algebras.- 5.6. Indecomposable and Irreducible Representations.- 6 Projective Modules over Artinian Algebras.- 6.1. Projective Modules.- 6.2. Homomorphisms of Projective Modules.- 6.3. Structure of Projective Modules.- 6.4. Idempotents.- 6.5. Structure of Artinian Algebras.- 6.6. Basic Algebras.- 6.7. Representation Type.- 7 Finite Representation Type.- 7.1. The Brauer-Thrall Conjectures.- 7.2. Bounded Representation Type.- 7.3. Sequence Categories.- 7.4. Simple Sequences.- 7.5. Almost Split Sequences.- 7.6. Almost Split Extensions.- 7.7. Roiter’s Theorem.- 8 Representation of Quivers.- 8.1. Constructing Modules.- 8.2. Representation of Quivers.- 8.3. Application to Algebras.- 8.4. Subquivers.- 8.5. Rigid Representations.- 8.6. Change of Orientation.- 8.7. Change of Representation.- 8.8. The Quadratic Space of a Quiver.- 8.9. Roots and Representations.- 9 Tensor Products.- 9.1. Tensor Products of R-modules.- 9.2. Tensor Products of Algebras.- 9.3. Tensor Products of Modules over Algebras.- 9.4. Scalar Extensions.- 9.5. Induced Modules.- 9.6. Morita Equivalence.- 10 Separable Algebras.- 10.1. Bimodules.- 10.2. Separability.- 10.3. Separable Algebras Are Finitely Generated.- 10.4. Categorical Properties.- 10.5. The Class of Separable Algebras.- 10.6. Extensions of Separable Algebras.- 10.7. Separable Algebras over Fields.- 10.8. Separable Extensions of Algebras.- 11 The Cohomology of Algebras.- 11.1. Hochschild Cohomology.- 11.2. Properties of Cohomology.- 11.3. The Snake Lemma.- 11.4. Dimension.- 11.5. Zero Dimensional Algebras.- 11.6. The Principal Theorem.- 11.7. Split Extensions of Algebras.- 11.8. Algebras with 2-nilpotent Radicals.- 12 Simple Algebras.- 12.1. Centers of Simple Algebras.- 12.2. The Density Theorem.- 12.3. The Jacobson-Bourbaki Theorem.- 12.4. Central Simple Algebras.- 12.5. The Brauer Group.- 12.6. The Noether-Skolem Theorem.- 12.7. The Double Centralizer Theorem.- 13 Subfields of Simple Algebras.- 13.1. Maximal Subfields.- 13.2. Splitting Fields.- 13.3. Algebraic Splitting Fields.- 13.4. The Schur Index.- 13.5. Separable Splitting Fields.- 13.6. The Cartan-Brauer-Hua Theorem.- 14 Galois Cohomology.- 14.1. Crossed Products.- 14.2. Cohomology and Brauer Groups.- 14.3. The Product Theorem.- 14.4. Exponents.- 14.5. Inflation.- 14.6. Direct Limits.- 14.7. Restriction.- 15 Cyclic Division Algebras.- 15.1. Cyclic Algebras.- 15.2. Constructing Cyclic Algebras by Inflation.- 15.3. The Primary Decomposition of Cyclic Algebras.- 15.4. Characterizing Cyclic Division Algebras.- 15.5. Division Algebras of Prime Degree.- 15.6. Division Algebras of Degree Three.- 15.7. A Non-cyclic Division Algebra.- 16 Norms.- 16.1. The Characteristic Polynomial.- 16.2. Computations.- 16.3. The Reduced Norm.- 16.4. Transvections and Dilatations.- 16.5. Non-commutative Determinants.- 16.6. The Reduced Whitehead Group.- 17 Division Algebras over Local Fields.- 17.1. Valuations of Division Algebras.- 17.2. Non-archimedean Valuations.- 17.3. Valuation Rings.- 17.4. The Topology of a Valuation.- 17.5. Local Fields.- 17.6. Extension of Valuations.- 17.7. Ramification.- 17.8. Unramified Extensions.- 17.9. Norm Factor Groups.- 17.10. Brauer Groups of Local Fields.- 18 Division Algebras over Number Fields.- 18.1. Field Composita.- 18.2. More Extensions of Valuations.- 18.3. Valuations of Algebraic Number Fields.- 18.4. The Albert-Hasse-Brauer-Noether Theorem.- 18.5. The Brauer Groups of Algebraic Number Fields.- 18.6. Cyclic Algebras over Number Fields.- 18.7. The Image of INV.- 19 Division Algebras over Transcendental Fields.- 19.1. The Norm Form.- 19.2. Quasi-algebraically Closed Fields.- 19.3. Krull’s Theorem.- 19.4. Tsen’s Theorem.- 19.5. The Structure of B(K(x)/F(x)).- 19.6. Exponents of Division Algebras.- 19.7. Twisted Laurent Series.- 19.8. Laurent Series Fields.- 19.9. Amitsur’s Example.- 20 Varieties of Algebras.- 20.1. Polynomial Identities and Varieties.- 20.2. Special Identities.- 20.3. Identities for Central Simple Algebras.- 20.4. Standard Identities.- 20.5. Generic Matrix Algebras.- 20.6. Central Polynomials.- 20.7. Structure Theorems.- 20.8. Universal Division Algebras.- References.- Index of Symbols.- Index of Terms.

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Book Synopsisto Computer Algebra.- Algebra of Polynomials, Rational Functions, and Power Series.- Normal Forms and Algebraic Representations.- Arithmetic of Polynomials, Rational Functions, and Power Series.- Homomorphisms and Chinese Remainder Algorithms.- Newton's Iteration and the Hensel Construction.- Polynomial GCD Computation.- Polynomial Factorization.- Solving Systems of Equations.- Gröbner Bases for Polynomial Ideals.- Integration of Rational Functions.- The Risch Integration Algorithm.Trade Review`The Computer Algebra community has been waiting for years for this book to appear. ...the book is a masterpiece and can be recommended to everyone interested in the algorithms for computer algebra, either as a reference for further research or just to give the casual user an idea why things work as good (or as bad) as they do in computer algebra packages. ...the recommendation would be clear: Buy this book! As it stands now my only advice is to stay away from this book, because you might be tempted to buy it anyway (at least I was).' Computer Algebra Nederland Newsletter, June 1993 Table of ContentsPreface. 1. Introduction to Computer Algebra. 2. Algebra of Polynomials, Rational Functions, and Power Series. 3. Normal Forms and Algebraic Representations. 4. Arithmetic of Polynomial, Rational Functions, and Power Series. 5. Homomorphisms and Chinese Remainder Algorithms. 6. Newton's Iteration and the Hensel Construction. 7. Polynomials GCD Computation. 8. Polynomial Factorization. 9. Solving Systems of Equation. 10. Grobner Bases for Polynomial Ideals. 11. Integration of Rational Functions. 12. The Risch Integration Algorithm. Notation. Index.

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Book SynopsisThis book explores some of the major turning points in the history of mathematics, ranging from ancient Greece to the present, demonstrating the drama that has often been a part of its evolution.Trade Review“This slim volume will be especially helpful to those teaching high school and junior college students the rudiments of selected ‘major turning points’ in the history of mathematics. It offers compact vignettes divided into ten basic topics: axiomatization, solutions of cubic equations, analytical geometry, probability, calculus, Gaussian integers, non-Euclidean geometries, hyper complex numbers, the infinite, and philosophy of mathematics from Hilbert to Gödel. … Thumbnail portraits and photographs of major mathematicians are also included.” (Joseph W. Dauben, Mathematical Reviews, May, 2017)“Turning Points does provide a useful summary and outline of at least a portion of the subject, and also functions nicely as a way of helping to mentally organize the material. It contains a number of good quotes, and a decent selection of bibliographic references at the end of each chapter. There are also problems at the end of each chapter, generally calling for essay-type answers that should require the student to do further reading.” (Mark Hunacek, MAA Reviews, maa.org, June, 2016)“Each chapter contains some problems and projects (they extend and increase the understanding of the material) as well as references and suggestions of further readings. A comprehensive index has been supplemented. The book can serve everyone interested in the historical development of mathematics – some mathematical background is of course required. It can serve teachers and students, can be used in courses in the history of mathematics as well as in courses in particular domains of mathematics … .” (Roman Murawski, zbMATH 1342.01005, 2016)Table of ContentsAxiomatics - Euclid's and Hilbert's: From Material to Formal.- Solution by Radicals of the Cubic: From Equations to Groups and from Real to Complex Numbers.- Analytic Geometry: From the Marriage of Two Fields to the Birth of a Third.- Probability: From Games of Chance to an Abstract Theory.- Calculus: From Tangents and Areas to Derivatives and Integrals.- Gaussian Integers: From Arithmetic to Arithmetics.- Non-Euclidean Geometry: From One Geometry to Many.- Hypercomplex Numbers: From Algebra to Algebras.- The Infinite: From Potential to Actual.- Philosophy of Mathematics: From Hilbert to Gödel.- Some Further Turning Points.- Index.

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Book SynopsisThisbookisintendedasanintroductiontoallthe?nitesimplegroups.During themonumentalstruggletoclassifythe?nitesimplegroups(andindeedsince), a huge amount of information about these groups has been accumulated. Conveyingthisinformationtothenextgenerationofstudentsandresearchers, not to mention those who might wish to apply this knowledge, has become a major challenge. With the publication of the two volumes by Aschbacher and Smith [12, 13] in 2004 we can reasonably regard the proof of the Classi?cation Theorem for Finite Simple Groups (usually abbreviated CFSG) as complete. Thus it is timely to attempt an overview of all the (non-abelian) ?nite simple groups in one volume. For expository purposes it is convenient to divide them into four basic types, namely the alternating, classical, exceptional and sporadic groups. The study of alternating groups soon develops into the theory of per- tation groups, which is well served by the classic text of Wielandt [170]and more modern treatments such as the comprehensive introduction by Dixon and Mortimer [53] and more specialised texts such as that of Cameron [19].Trade ReviewFrom the reviews: “The book under review has as its main goal to give an introductory overview of the construction and main properties of all finite simple groups. … This book is the first one that attempts to give a systematic treatment of all finite simple groups, using the more recent and efficient constructions … . The author succeeds in making this important but difficult area of mathematics readily accessible to a large sector of the mathematical community, and for this we should be grateful.” (Felipe Zaldivar, The Mathematical Association of America, March, 2010)“One of the great achievements of mathematics was the classification of the finite simple groups … . the book brings much information to the classroom. It contains exactly those things one would like to know if one were to meet the individual simple groups for the first time. … perfectly suitable for an advanced course or seminar. … accessible also to those who are not great experts in group theory. Anyone interested in finite groups … should have this book on his or her bookshelf.” (Gernot Stroth, Mathematical Reviews, Issue 2011 e)“The author of this book has succeeded in giving an overview of all non-abelian finite simple groups which is accessible to non-experts. … For anyone who wants to get information on finite simple groups without having to tackle massive monographs this volume will be most welcome.” (Ch. Baxa, Monatshefte für Mathematik, Vol. 164 (3), November, 2011)“It is the first text at this level in which all the finite simple groups are treated together, pointing out their connections. … The text is very well organised. The introduction, which forms the first chapter, contains a brief history and the statement of the classification theorem, together with sections giving remarks on the applications and the proof of the theorem. … Consequently the book may also be useful to a reader who just wants an introduction to a particular group or family of groups.” (Peter Shiu, The Mathematical Gazette, Vol. 95 (532), March, 2011)“This book is a unique introductory overview of all the finite simple groups, and thus it is suitable not only for specialists who are interested in finite simple groups but also for advanced undergraduate and graduate students in algebra. The section entitled ‘Further reading’ at the end of each chapter is a nice guide to further study of the subjects.” (Hiromichi Yamada, Zentralblatt MATH, Vol. 1203, 2011)Table of ContentsThe alternating groups.- The classical groups.- The exceptional groups.- The sporadic groups.

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Book SynopsisA modern and student-friendly introduction to this popular subject: it takes a more "natural" approach and develops the theory at a gentle pace with an emphasis on clear explanations Features plenty of worked examples and exercises, complete with full solutions, to encourage independent study Previous books by Howie in the SUMS series have attracted excellent reviews Trade ReviewFrom the reviews:“This is a short but very good introductory book on abstract algebra, with emphasis on Galois Theory. Very little background in mathematics is required, so that the potential audience for this book range from undergraduate and graduate students, researchers, computer professionals, and the math enthusiasts.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, July, 2013)"The author wrote this book to provide the reader with a treatment of classical Galois theory. … The book is well written. It contains many examples and over 100 exercises with solutions in the back of the book. Sprinkled throughout the book are interesting commentaries and historical comments. The book is suitable as a textbook for upper level undergraduate or beginning graduate students." (John N. Mordeson, Zentralblatt MATH, Vol. 1103 (5), 2007)"To write such a book on a widely known but genuinely non-trivial topic is a challenge. … J. M. Howie did exactly what it takes. And he did it with such vigour and skill that the outcome is indeed absorbing and astounding. … Every paragraph has been scheduled with utmost care and the proofs are crystal clear. … the reader will never feel forlorn amidst brilliant theorems, which makes the book such a good read." (J. Lang, Internationale Mathematische Nachrichten, Issue 206, 2007)"Howie’s book ... provides a rigorous and thorough introduction to Galois theory. ... this book would be an excellent choice for anyone with at least some backgound in abstract algebra who seeks an introduction to the study of Galois theory. Summing Up: Highly recommended. Upper-division undergraduates; graduate students." (D. S. Larson, CHOICE, Vol. 43 (10), June, 2006)"The latest addition to Springer’s Undergraduate Mathematics Series is John Howie’s Fields and Galois Theory. … Howie is a fine writer, and the book is very self-contained. … I know that many of my students would appreciate Howie’s approach much more as it is not as overwhelming. This book also has a large number of good exercises, all of which have solutions in the back of the book. All in all, Howie has done a fine job writing a book on field theory … ." (Darren Glass, MathDL, February, 2006)"The book can serve as a useful introduction to the theory of fields and their extensions. The relevant background material on groups and rings is covered. The text is interspersed with many worked examples, as well as more than 100 exercises, for which solutions are provided at the end." (Chandan Singh Dalawat, Mathematical Reviews, Issue 2006 g)Table of ContentsRings and Fields.- Integral Domains and Polynomials.- Field Extensions.- Applications to Geometry.- Splitting Fields.- Finite Fields.- The Galois Group.- Equations and Groups.- Some Group Theory.- Groups and Equations.- Regular Polygons.- Solutions.

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Book SynopsisThis book provides a comprehensive, in-depth overview of elementary mathematics as explored in Mathematical Olympiads around the world. It expands on topics usually encountered in high school and could even be used as preparation for a first-semester undergraduate course. This third and last volume covers Counting, Generating Functions, Graph Theory, Number Theory, Complex Numbers, Polynomials, and much more.As part of a collection, the book differs from other publications in this field by not being a mere selection of questions or a set of tips and tricks that applies to specific problems. It starts from the most basic theoretical principles, without being either too general or too axiomatic. Examples and problems are discussed only if they are helpful as applications of the theory. Propositions are proved in detail and subsequently applied to Olympic problems or to other problems at the Olympic level.The book also explores some of the hardest problems presented at National and International Mathematics Olympiads, as well as many essential theorems related to the content. An extensive Appendix offering hints on or full solutions for all difficult problems rounds out the book.Table of Contents

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Book SynopsisThis English edition of Yuri I. Manin's well-received lecture notes provides a concise but extremely lucid exposition of the basics of algebraic geometry and sheaf theory. The lectures were originally held in Moscow in the late 1960s, and the corresponding preprints were widely circulated among Russian mathematicians. This book will be of interest to students majoring in algebraic geometry and theoretical physics (high energy physics, solid body, astrophysics) as well as to researchers and scholars in these areas."This is an excellent introduction to the basics of Grothendieck's theory of schemes; the very best first reading about the subject that I am aware of. I would heartily recommend every grad student who wants to study algebraic geometry to read it prior to reading more advanced textbooks."- Alexander BeilinsonTrade Review“This slim volume is still a valuable introduction to schemes and nicely complements the textbooks on this topic which have appeared in the meantime.” (C. Baxa, Monatshefte für Mathematik, Vol. 201 (4), August, 2023)“Throughout the text there are many instructive examples, remarks, and clarifying footnotes. The style of exposition is rather concise, very elegant, extremely lucid and enlightening, versatile, and – despite its venerable age of fifty years – absolutely modern and timely. … an excellent source for students, instructors, and mathematical physicists. No doubt, with this textbook, the mathematical community has another general standard reference in algebraic geometry at its disposal.” (Werner Kleinert, zbMATH 1390.14002, 2018)Table of ContentsEditor's Preface.- Author's Preface.- 1 Affine Schemes.- 2 Sheaves, Schemes, and Projective Spaces.- References.- Index.

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Book SynopsisGalois theory has such close analogies with the theory of coverings that algebraists use a geometric language to speak of field extensions, while topologists speak of "Galois coverings". This book endeavors to develop these theories in a parallel way, starting with that of coverings, which better allows the reader to make images. The authors chose a plan that emphasizes this parallelism. The intention is to allow to transfer to the algebraic framework of Galois theory the geometric intuition that one can have in the context of coverings. This book is aimed at graduate students and mathematicians curious about a non-exclusively algebraic view of Galois theory.Trade Review“This book covers a lot of interesting material and is surely a valuable addition to the literature, but is certainly not for the timid. It brings together a broad array of sophisticated mathematics … and it does so in a very general and abstract way, with an exposition that gives whole new meaning to the word ‘concise’.” (Mark Hunacek, MAA Reviews, April 5, 2021)Table of ContentsIntroduction.- Chapter 1. Zorn’s Lemma.- Chapter 2. Categories and Functors.- Chapter 3. Linear Algebra.- Chapter 4. Coverings.- Chapter 5. Galois Theory.- Chapter 6. Riemann Surfaces.- Chapter 7. Dessins d’Enfants.- Bibliography.- Index of Notation

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