Algebra Books

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Book SynopsisFollowing a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields.Trade ReviewFrom the reviews of the second edition:"This well-written book covers the basic facts about the geometry and arithmetic of elliptic curves, and is sure to become the standard reference in the subject. It meets the needs of at least three groups of people: students interested in doing research in Diophantine geometry, mathematicians needing a reference for standard facts about elliptic curves, and computer scientists interested in algorithms and needing an introduction to elliptic curves..."-- MATHEMATICAL REVIEWS“The book under review is the second, revised, enlarged, and updated edition of J. Silverman’s meanwhile classical primer of the arithmetic of elliptic curves. … All together, this enlarged and updated version of J. Silverman’s classic ‘The Arithmetic of Elliptic Curves’ significantly increases the unchallenged value of this modern primer as a standard textbook in the field. … This makes the entire text a perfect source for teachers and students, for courses and self-study, and for further studies in the arithmetic of elliptic curves likewise.” (Werner Kleinert, Zentralblatt MATH, Vol. 1194, 2010)“For the second edition of his masterly book, the author considerably updated and improved several results and proofs. … book contains a great many exercises, many of which develop or complement the results from the main body of the book. … The reference list contains 317 items and reflects both classical and recent achievements on the topic. Notes on the exercises are an aid to the reader. … Summarizing, this is an excellent book … . useful both for experienced mathematicians and for graduate students.”­­­ (Vasil' I. Andriĭchuk, Mathematical Reviews, Issue 2010 i)“This is the second edition of an excellent textbook on the arithmetical theory of elliptic curves … . Although there are now a number of good books on this topic it has stood the test of time and become a popular introductory text and a standard reference. … The author has added remarks which point out their significance and connection to those parts of the theory he presents. These will give readers a good start if they want to study one of them.” (Ch. Baxa, Monatshefte für Mathematik, Vol. 164 (3), November, 2011)“The book is written for graduate students … and for researchers interested in standard facts about elliptic curves. … A wonderful textbook on the arithmetic theory of elliptic curves and it is a very popular introduction to the subject. … I recommend this book for anyone interested in the mathematical study of elliptic curves. It is an excellent introduction, elegant and very well written. It is one of the best textbooks to graduate level studies I have ever had contact yet.” (Book Inspections Blog, 2012)Table of ContentsAlgebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of an Elliptic Curve.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.- Computing the Mordell#x2013;Weil Group.- Algorithmic Aspects of Elliptic Curves.

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Book SynopsisThe Integers.- Groups.- Rings.- Polynomials.- Vector Spaces and Modules.- Some Linear Groups.- Field Theory.- Finite Fields.- The Real and Complex Numbers.- Sets.Trade ReviewFrom the reviews of the third edition: "As is very typical for Professor Lang’s self demand and style of publishing, he has tried to both improve and up-date his already well-established text. … Numerous examples and exercises accompany this now already classic primer of modern algebra, which as usual, reflects the author’s great individuality just as much as his unrivalled didactic mastery and his care for profound mathematical education at any level. … The present textbook … will remain one of the great standard introductions to the subject for beginners." (Werner Kleinert, Zentralblatt MATH, Vol. 1063, 2005)Table of Contents* Foreword * The Integers * Groups * Rings * Polynomials * Vector Spaces and Modules * Some Linear Groups * Field Theory * Finite Fields * The Real and Complex Numbers * Sets * Appendix * Index

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Book SynopsisFirst textbook-level account of basic examples and techniques in this area.Suitable for self-study by a reader who knows a little commutative algebra and algebraic geometry already.Trade ReviewFrom the reviews:"This book is devoted to offer … an approach to the study of this algebraic subject (syzygy = relation among generators of a module) … . a student would learn a lot of algebraic geometry from it. The double bet of the book is to be able to be a complete textbook … and at the same time to become a useful reference text for research work on the subject. I would say that both aspects of the bet have been gained … ." (Alessandro Gimigliano, Zentralblatt MATH, Vol. 1066, 2005)"This book may be regarded as a complement to the author’s Commutative Algebra … . It begins by explaining syzygies and their connection with the Hilbert function, and turns to describing various aspects of algebraic geometry … . Two appendices provide the background in commutative algebra and local cohomology. Together with exercises, it gives a good survey of topics often not covered." (Mathematika, Vol. 52, 2005)"This monograph is devoted to the geometric properties of a projective variety corresponding to the properties of its syzygies … . Altogether, this is a most welcome addition to the literature and will help many a reader bridge the gap between the abstractions of algebra and the more tangible field of geometry." (Ch. Baxa, Monatshefte für Mathematik, Vol. 150 (1), 2006)Aus den Rezensionen: “... Das vorliegende Buch beschäftigt sich mit der qualitativen geometrischen Theorie der Syzygien. ... Es gibt zwei sehr kompakt geschriebene Anhänge: Der erste führt in die lokale Kohomologie ein, der zweite stellt für das Buch nötige Vorkenntnisse der kommutativen Algebra ... zusammen. Dieses Buch ist sehr elegant geschrieben und vermittelt viele interessante Ideen. In der Lehre könnte es gut für weiterführende Vorlesungen über algebraische Geometrie verwendet werden.“ (Franz Pauer, in: Internationale Mathematische Nachrichten, December/2009, Issue 12, S. 45)“This very interesting book is the first textbook-level account of syzygies as they are used in algebraic geometry. … The reader will find two very good and useful appendices. … The book can be read, without any problem, by a student who has received already a little introduction in commutative algebra and algebraic geometry. I highly recommend this nice and deep textbook for all students and researchers studying algebraic geometry or commutative algebra.” (Dominique Lambert, Bulletin of the Belgian Mathematical Society, Vol. 15 (1), 2008)Table of ContentsPreface: Algebra and Geometry * Free Resolutions and Hilbert Functions * First Examples of Free Resolutions * Points in P^2 * Castelnuovo—Mumford Regularity * The Regularity of Projective Curves * Linear Series and 1-Generic Matrices * Linear Complexes and the Linear Syzygy Theorem * Curves of High Degree * Clifford Index and Canonical Embedding * Appendix 1: Introduction to Local Cohomology * Appendix 2: A Jog Through Commutative Algebra * References * Index

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Book SynopsisThis book deals with several aspects of what is now called "explicit number theory." The local aspect, global aspect, and the third aspect is the theory of zeta and L-functions.Trade ReviewFrom the reviews:"Cohen (Université Bordeaux I, France), an instant classic, uniquely bridges the gap between old-fashioned, naive treatments and the many modern books available that develop the tools just mentioned … . Summing Up: Recommended. … Upper-division undergraduates through faculty." (D. V. Feldman, CHOICE, Vol. 45 (5), January, 2008)"The book deals with aspects of ‘explicit number theory’. … The central theme … is the solution of Diophantine equations. … It combines an interesting ‘philosophy’ of the subject with an encyclopedic grasp of detail. The extension of the author’s reach via the contributed chapters is a good idea. Perhaps it is the start of a trend, as the subject grows more and more. … It will undoubtedly be mined by instructors for their graduate courses, particularly for the purpose of including some recently-proved content." (R. C. Baker, Mathematical Reviews, Issue 2008 e)“This is the second volume of a highly impressive two-volume textbook on Diophantine analysis. … readers are presented with an almost overwhelming amount of material. This … text book is bound to become an important reference for students and researchers alike.” (C. Baxa, Monatshefte für Mathematik, Vol. 157 (2), June, 2009)Table of ContentsAnalytic Tools.- Bernoulli Polynomials and the Gamma Function.- Dirichlet Series and L-Functions.- p-adic Gamma and L-Functions.- Modern Tools.- Applications of Linear Forms in Logarithms.- Rational Points on Higher-Genus Curves.- The Super-Fermat Equation.- The Modular Approach to Diophantine Equations.- Catalan’s Equation.

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Book SynopsisUniversal Algebra has become the most authoritative, consistently relied on text in a field with applications in other branches of algebra and other fields such as combinatorics, geometry, and computer science.Each chapter is followed by an extensive list of exercises and problems.Table of ContentsBasic Concepts.- Subalgebras and Homomorphisms.- Partial Algebras.- Constructions of Algebras.- Free Algebras.- Independence.- Elements of Model Theory.- Elementary Properties of Algebraic Constructions.- Free Structures.- Appendix 1. General Survey.- Appendix 2. The Problems.- Appendix 3. Congruence Varieties.- Appendix 4. Equational Logic by Walter Taylor.- Appendix 5. Primality the influence of Boolean Algebras in Universal Algebra.- Appendix 6 Equational Compactness.- Appendix 7 The Independence Proof.

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Book SynopsisThis book is a significant reworking of an earlier work by the authors. It offers a comprehensive introduction to Lie theory, representation theory, invariant theory, and algebraic groups, in a new presentation that includes a broader range of applications.Trade ReviewFrom the reviews:"This is a terrific book, succeeding in its considerable ambitions, and doing so in the remarkable style … . Symmetry, Representations, and Invariants is indeed capable of instructing the reader in the three themes given in its title, taking him from elementary and foundational notions to very advanced material … . Symmetry, Representations, and Invariants promises to be a graduate text of major importance." (Michael Berg, The Mathematical Association of America, August, 2009)“The book under review is a comprehensive introduction to Lie theory, representation theory, invariant theory, and algebraic groups. … can be used as a source for various kinds of courses. … supported by the rich collections of exercises (mostly with detailed hints for solutions) accompanying each section. Local reading is well supported by the structure of the book. The book can be recommended for a … wide audience of readers: for graduate and postgraduate students as well as for researchers as a reference work.” (Sergei Platonov, Zentralblatt MATH, Vol. 1173, 2009)“The book under review is a substantial rewriting and extension of the authors’ previous work ‘Representations and Invariants of the Classical Groups’. It is a thorough and comprehensive treatment of the geometry of Lie and algebraic group actions. … More than 350 exercises are presented. This monograph is written with great care and provides an excellent basis for a variety of courses from this field.” (M. Kunzinger, Monatshefte für Mathematik, Vol. 163 (1), May, 2011)“The book will serve as a useful reference for a broad range of mathematicians, as well as physicists seeking a rigorous, systematic development of the representation theory underlying much of modern quantum mechanics. Detailed and careful statements of definitions, theorems, and proofs are provided, as well as over 350 exercises, many of them substantial. … can be used as the basis for graduate-level courses in Lie groups and algebraic groups, representation theory, invariant theory, and a variety of applications.” (Peter J. Olver, SIAM Review, Vol. 53 (3), 2011)“This volume is clearly a labour of love on the part of the authors, who have obviously thought very carefully about the best way to explain and motivate this often sophisticated material. … The authors have taken great pains to make this book a textbook as well as a useful reference. … It should be in the library of every university with a graduate mathematics program, as well as on the shelf of every lecturer teaching, or with research interests in, the material covered here.” (Mark Hunacek, The Mathematical Gazette, Vol. 96 (536), July, 2012)Table of ContentsLie Groups and Algebraic Groups.- Structure of Classical Groups.- Highest-Weight Theory.- Algebras and Representations.- Classical Invariant Theory.- Spinors.- Character Formulas.- Branching Laws.- Tensor Representations of GL(V).- Tensor Representations of O(V) and Sp(V).- Algebraic Groups and Homogeneous Spaces.- Representations on Spaces of Regular Functions.

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Book SynopsisThis book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations.Table of ContentsI. Basic Concepts.- 1. Definitions and first examples.- 1.1 The notion of Lie algebra.- 1.2 Linear Lie algebras.- 1.3 Lie algebras of derivations.- 1.4 Abstract Lie algebras.- 2. Ideals and homomorphisms.- 2.1 Ideals.- 2.2 Homomorphisms and representations.- 2.3 Automorphisms.- 3. Solvable and nilpotent Lie algebras.- 3.1 Solvability.- 3.2 Nilpotency.- 3.3 Proof of Engel’s Theorem.- II. Semisimple Lie Algebras.- 4. Theorems of Lie and Cartan.- 4.1 Lie’s Theorem.- 4.2 Jordan-Chevalley decomposition.- 4.3 Cartan’s Criterion.- 5. Killing form.- 5.1 Criterion for semisimplicity.- 5.2 Simple ideals of L.- 5.3 Inner derivations.- 5.4 Abstract Jordan decomposition.- 6. Complete reducibility of representations.- 6.1 Modules.- 6.2 Casimir element of a representation.- 6.3 Weyl’s Theorem.- 6.4 Preservation of Jordan decomposition.- 7. Representations of sl (2, F).- 7.1 Weights and maximal vectors.- 7.2 Classification of irreducible modules.- 8. Root space decomposition.- 8.1 Maximal toral subalgebras and roots.- 8.2 Centralizer of H.- 8.3 Orthogonality properties.- 8.4 Integrality properties.- 8.5 Rationality properties Summary.- III. Root Systems.- 9. Axiomatics.- 9.1 Reflections in a euclidean space.- 9.2 Root systems.- 9.3 Examples.- 9.4 Pairs of roots.- 10. Simple roots and Weyl group.- 10.1 Bases and Weyl chambers.- 10.2 Lemmas on simple roots.- 10.3 The Weyl group.- 10.4 Irreducible root systems.- 11. Classification.- 11.1 Cartan matrix of ?.- 11.2 Coxeter graphs and Dynkin diagrams.- 11.3 Irreducible components.- 11.4 Classification theorem.- 12. Construction of root systems and automorphisms.- 12.1 Construction of types A-G.- 12.2 Automorphisms of ?.- 13. Abstract theory of weights.- 13.1 Weights.- 13.2 Dominant weights.- 13.3 The weight ?.- 13.4 Saturated sets of weights.- IV. Isomorphism and Conjugacy Theorems.- 14. Isomorphism theorem.- 14.1 Reduction to the simple case.- 14.2 Isomorphism theorem.- 14.3 Automorphisms.- 15. Cartan subalgebras.- 15.1 Decomposition of L relative to ad x.- 15.2 Engel subalgebras.- 15.3 Cartan subalgebras.- 15.4 Functorial properties.- 16. Conjugacy theorems.- 16.1 The group g (L).- 16.2 Conjugacy of CSA’s (solvable case).- 16.3 Borel subalgebras.- 16.4 Conjugacy of Borel subalgebras.- 16.5 Automorphism groups.- V. Existence Theorem.- 17. Universal enveloping algebras.- 17.1 Tensor and symmetric algebras.- 17.2 Construction of U(L).- 17.3 PBW Theorem and consequences.- 17.4 Proof of PBW Theorem.- 17.5 Free Lie algebras.- 17. Generators and relations.- 17.1 Relations satisfied by L.- 17.2 Consequences of (S1)-(S3).- 17.3 Serre’s Theorem.- 17.4 Application: Existence and uniqueness theorems.- 18. The simple algebras.- 18.1 Criterion for semisimplicity.- 18.2 The classical algebras.- 18.3 The algebra G2.- VI. Representation Theory.- 20. Weights and maximal vectors.- 20.1 Weight spaces.- 20.2 Standard cyclic modules.- 20.3 Existence and uniqueness theorems.- 21. Finite dimensional modules.- 21.1 Necessary condition for finite dimension.- 21.2 Sufficient condition for finite dimension.- 21.3 Weight strings and weight diagrams.- 21.4 Generators and relations for V(?).- 22. Multiplicity formula.- 22.1 A universal Casimir element.- 22.2 Traces on weight spaces.- 22.3 Freudenthal’s formula.- 22.4 Examples.- 22.5 Formal characters.- 23. Characters.- 23.1 Invariant polynomial functions.- 23.2 Standard cyclic modules and characters.- 23.3 Harish-Chandra’s Theorem.- 24. Formulas of Weyl, Kostant, and Steinberg.- 24.1 Some functions on H*.- 24.2 Kostant’s multiplicity formula.- 24.3 Weyl’s formulas.- 24.4 Steinberg’s formula.- VII. Chevalley Algebras and Groups.- 25. Chevalley basis of L.- 25.1 Pairs of roots.- 25.2 Existence of a Chevalley basis.- 25.3 Uniqueness questions.- 25.4 Reduction modulo a prime.- 25.5 Construction of Chevalley groups (adjoint type).- 26. Kostant’s Theorem.- 26.1 A combinatorial lemma.- 26.2 Special case: sl (2, F).- 26.3 Lemmas on commutation.- 26.4 Proof of Kostant’s Theorem.- 27. Admissible lattices.- 27.1 Existence of admissible lattices.- 27.2 Stabilizer of an admissible lattice.- 27.3 Variation of admissible lattice.- 27.4 Passage to an arbitrary field.- 27.5 Survey of related results.- References.- Afterword (1994).- Index of Terminology.- Index of Symbols.

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Book SynopsisThe parts of the book concerned with this aspect of the subject are Chapters I, IV, and V dealing respectively with finite dimen­ sional field extensions and Galois theory, general structure theory of fields, and valuation theory.Table of Contents1. Extension of homomorphisms.- 2. Algebras.- 3. Tensor products of vector spaces.- 4. Tensor product of algebras.- I: Finite Dimensional Extension Fields.- 1. Some vector spaces associated with mappings of fields.- 2. The Jacobson-Bourbaki correspondence.- 3. Dedekind independence theorem for isomorphisms of a field.- 4. Finite groups of automorphisms.- 5. Splitting field of a polynomial.- 6. Multiple roots. Separable polynomials.- 7. The “fundamental theorem” of Galois theory.- 8. Normal extensions. Normal closures.- 9. Structure of algebraic extensions. Separability.- 10. Degrees of separability and inseparability. Structure of normal extensions.- 11. Primitive elements.- 12. Normal bases.- 13 Finite fields.- 14. Regular representation, trace and norm.- 15. Galois cohomology.- 16 Composites of fields.- II: Galois Theory of Equations.- 1. The Galois group of an equation.- 2. Pure equations.- 3. Galois’ criterion for solvability by radicals.- 4. The general equation of n-th degree.- 5. Equations with rational coefficients and symmetric group as Galois group.- III: Abelian Extensions.- 1. Cyclotomic fields over the rationals.- 2. Characters of finite commutative groups.- 3. Kummer extensions.- 4. Witt vectors.- 5. Abelian p-extensions.- IV: Structure Theory of Fields.- 1. Algebraically closed fields.- 2. Infinite Galois theory.- 3. Transcendency basis.- 4. Lüroth’s theorem.- 5. Linear disjointness and separating transcendency bases.- 6. Derivations.- 7. Derivations, separability and p-independence.- 8. Galois theory for purely inseparable extensions of exponent one.- 9. Higher derivations.- 10. Tensor products of fields.- 11. Free composites of fields.- V: Valuation Theory.- 1. Real valuations.- 2. Real valuations of the field of rational numbers.- 3. Real valuations of ?(x) which are trivial in ?.- 4. Completion of a field.- 5. Some properties of the field of p-adic numbers.- 6. Hensel’s lemma.- 7. Construction of complete fields with given residue fields.- 8. Ordered groups and valuations.- 9. Valuations, valuation rings, and places.- 10. Characterization of real non-archimedean valuations.- 11. Extension of homomorphisms and valuations.- 12. Application of the extension theorem: Hilbert Nullstellensatz.- 13. Application of the extension theorem: integral closure.- 14. Finite dimensional extensions of complete fields.- 15. Extension of real valuations to finite dimensional extension fields.- 16. Ramification index and residue degree.- VI: Artin-Schreier Theory.- 1. Ordered fields and formally real fields.- 2. Real closed fields.- 3. Sturm’s theorem.- 4. Real closure of an ordered field.- 5. Real algebraic numbers.- 6. Positive definite rational functions.- 7. Formalization of Sturm’s theorem. Resultants.- 8. Decision method for an algebraic curve.- 9. Equations with parameters.- 10. Generalized Sturm’s theorem. Applications.- 11. Artin-Schreier characterization of real closed fields.- Suggestions for further reading.

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Book Synopsis1 WedderburnArtin Theory.- 2 Jacobson Radical Theory.- 3 Introduction to Representation Theory.- 4 Prime and Primitive Rings.- 5 Introduction to Division Rings.- 6 Ordered Structures in Rings.- 7 Local Rings, Semilocal Rings, and Idempotents.- 8 Perfect and Semiperfect Rings.- References.- Name Index.Trade ReviewFrom the reviews of the second edition: "Ten years ago, the first edition ... of this book appeared. It is quite rare that a book can become a classic in such a short time, but this did happen for this excellent book. Of course minor changes were made for the second edition; new exercises and an appendix on uniserial modules were added. Every part of the text was written with love and care. The explanations are very well done, useful examples help to understand the material ... ." (G. Pilz, Internationale Mathematische Nachrichten, Issue 196, 2004) "This is the second edition of the book with the same title published in 1991. The entire text has been retyped, some proofs were rewritten, and numerous improvements in the exposition have been included. ... The original exercises in the first edition have been replaced by 400 exercises of the Problem Book published by the author in 1995 ... . As the publication ... impressively shows, the book is indeed very suitable as a text for a full-year graduate course in non-commutative ring theory." (H. Mitsch, Monatshefte für Mathematik, Vol. 136 (4), 2002) "The book under review is the new edition of the 1st edition of this book which was published ten years ago. In the intervening period, the book has become a standard textbook for a graduate course in non commutative rings. ... All this material is beautifully covered in this book with many examples and exercises. The book is a joy to read, and must be read by all students of mathematics. It is specially welcome to have this book in softcover, relatively in-expensive, edition ... ." (Dipendra Prasad, Journal of the Indian Academy of Mathematics, Vol. 24 (1), 2002) "This book, an outgrowth of the author’s lectures at the University of Berkeley, is intended as a textbook for a one-semester course in basic ring theory. ... More than 400 exercises testing the understanding of the general theory in the text are included in this new edition." (L’Enseignement Mathematique, Vol. 47 (3-4), 2001) "The present book is a radical update. For the second edition the text was retyped, some proofs were rewritten and improvements in exposition have also taken place. ... It is well-written and consists of eight chapters. ... There is a very good reference section for further study and a name index consisting of four pages of closely-packed names. ... As always the standard of print and presentation by Springer is exemplary." (Brian Denton, The Mathematical Gazette, Vol. 86 (505), 2002)Table of Contents* Wedderburn-Artin Theory * Jacobson Radical Theory * Introduction to Representation Theory * Prime and Primitive Rings * Introduction to Division Rings * Local Rings, Semilocal Rings, and Idempotents * Perfect and Semiperfect Rings * References

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Book SynopsisCongruent Numbers in General.- Congruences of the First Degree.- Residues of Powers.- Congruences of the Second Degree.- Forms and Indeterminate Equations of the Second Degree.- Various Applications of the Preceding Discussions.- Equations Defining Sections of a Circle.- Additional Notes.- Tables.- Gauss' Handwritten Notes.- List of Special Symbols.- Directory of Terms.Table of ContentsCongruent Numbers in General.- Congruences of the First Degree.- Residues of Powers.- Congruences of the Second Degree.- Forms and Indeterminate Equations of the Second Degree.- Various Applications of the Preceding Discussions.- Equations Defining Sections of a Circle.- Additional Notes.- Tables.- Gauss' Handwritten Notes.- List of Special Symbols.- Directory of Terms.

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Book SynopsisPerfect for history buffs and armchair algebra experts, Unknown Quantity tells the story of the development of abstract mathematical thought.John Derbyshire discovers the story behind the formulae, roots, and radicals. As he did so masterfully in Prime Obsession, Derbyshire brings the evolution of mathematical thinking to dramatic life by focusing on the key historical players. Unknown Quantity begins in the time of Abraham and Isaac and moves from Abel's proof to the higher levels of abstraction developed by Galois through modern-day advances. Derbyshire explains how a simple turn of thought from this plus this equals this to this plus what equals this? gave birth to a whole new way of perceiving the world. With a historian's narrative authority and a beloved teacher's clarity and passion, Derbyshire leads readers on an intellectually satisfying and pleasantly challenging historical and mathematical journey.

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