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Algebra de B. L. Van Der Waerden

Algebra

Autor B. L. Van Der Waerden Traducere de J. R. Schulenberger
en Limba Engleză Paperback – 17 oct 2003

În această ediție, regăsim prima tipărire în format paperback a traducerii consacrate în limba engleză, păstrând rigoarea și stilul auster care au definit predarea algebrei abstracte la nivel universitar. Volumul de față consolidează materialul prezentat anterior, extinzând cadrul teoretic prin capitole dedicate algebrelor, inelelor cu condiție minimală și descompunerilor centrale. Descoperim aici o sinteză magistrală între originalitatea conceptuală a lui Emmy Noether și claritatea expunerii lui Emil Artin, elemente pe care B. L. Van Der Waerden le-a integrat încă din prima ediție germană a lucrării sale.

Structura volumului reflectă o progresie logică spre structuri tot mai complexe. Primele secțiuni din acest volum (capitolele 12 și 13) se concentrează pe algebra liniară, explorând modulele peste inele și formele bilineare antisimetrice, pentru ca ulterior să detalieze teoria reprezentării. Această abordare unificată a structurilor algebrice extinde cadrul propus de Algebraic Structures de George R. Kempf cu date noi și o rigoare axiomatică ce a transformat disciplina dintr-o colecție de tehnici de calcul într-o știință a structurilor. Dacă Basic Algebra de P. M. Cohn este adesea preferat pentru o introducere modernă și modulară, textul lui B. L. Van Der Waerden rămâne sursa primară pentru înțelegerea fundamentelor, păstrându-și relevanța în curriculumul de licență.

Poziționată în continuarea lucrării Algebra I, această carte reprezintă maturizarea viziunii autorului asupra geometriei algebrice și a metodelor abstracte. Remarcăm faptul că, deși stilul este unul concis, progresia de la divizori elementari la teoremele de structură pentru inele semisimple oferă cititorului o imagine completă asupra algebrei moderne, așa cum a fost ea cristalizată în secolul trecut.

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Specificații

ISBN-13: 9780387406251
ISBN-10: 0387406255
Pagini: 300
Ilustrații: XII, 284 p. 1 illus.
Dimensiuni: 155 x 235 x 17 mm
Greutate: 0.46 kg
Ediția:1st ed. 1991. 1st softcover printing 2003
Editura: Springer
Locul publicării:New York, NY, United States

Public țintă

Lower undergraduate

De ce să citești această carte

Această ediție este esențială pentru studenții la matematică și cercetătorii care doresc să studieze algebra abstractă direct de la sursa care a definit disciplina. Cititorul câștigă o înțelegere profundă a structurilor axiomatice, beneficiind de sinteza istorică dintre ideile lui Noether și Artin. Este o resursă de referință pentru cursuri de licență și master, oferind un model de claritate și rigoare matematică greu de egalat.

Descriere scurtă

...This beautiful and eloquent text served to transform the graduate teaching of algebra, not only in Germany, but elsewhere in Europe and the United States. It formulated clearly and succinctly the conceptual and structural insights which Noether had expressed so forcefully. This was combined with the elegance and understanding with which Artin had lectured...Its simple but austere style set the pattern for mathematical texts in other subjects, from Banach spaces to topological group theory...It is, in my view, the most influential text in algebra of the twentieth century.
- Saunders MacLane, Notices of the AMS
How exciting it must have been to hear Emil Artin and Emmy Noether lecture on algebra in the 1920's, when the axiomatic approach to the subject was amazing and new! Van der Waerden was there, and produced from his notes the classic textbook of the field. To Artin's clarity and Noether's originality he added his extraordinary gift for synthesis. At one time every would-be algebraist had to study this text. Even today, all who work in Algebra owe a tremendous debt to it; they learned from it by second or third hand, if not directly. It is still a first-rate (some would say, the best) source for the great range of material it contains.
- David Eisenbud, Mathematical Sciences Research Institute
Van der Waerden's book Moderne Algebra, first published in 1930, set the standard for the unified approach to algebraic structures in the twentieth century. It is a classic, still worth reading today.
- Robin Hartshorne, University of California, Berkeley

Cuprins

12 Linear Algebra.- 12.1 Modules over a Ring.- 12.2 Modules over Euclidean Rings. Elementary Divisors.- 12.3 The Fundamental Theorem of Abelian Groups.- 12.4 Representations and Representation Modules.- 12.5 Normal Forms of a Matrix in a Commutative Field.- 12.6 Elementary Divisors and Characteristic Functions.- 12.7 Quadratic and Hermitian Forms.- 12.8 Antisymmetric Bilinear Forms.- 13 Algebras.- 13.1 Direct Sums and Intersections.- 13.2 Examples of Algebras.- 13.3 Products and Crossed Products.- 13.4 Algebras as Groups with Operators. Modules and Representations.- 13.5 The Large and Small Radicals.- 13.6 The Star Product.- 13.7 Rings with Minimal Condition.- 13.8 Two-Sided Decompositions and Center Decomposition.- 13.9 Simple and Primitive Rings.- 13.10 The Endomorphism Ring of a Direct Sum.- 13.11 Structure Theorems for Semisimple and Simple Rings.- 13.12 The Behavior of Algebras under Extension of the Base Field.- 14 Representation Theory of Groups and Algebras.- 14.1 Statement of the Problem.- 14.2 Representation of Algebras.- 14.3 Representations of the Center.- 14.4 Traces and Characters.- 14.5 Representations of Finite Groups.- 14.6 Group Characters.- 14.7 The Representations of the Symmetric Groups.- 14.8 Semigroups of Linear Transformations.- 14.9 Double Modules and Products of Algebras.- 14.10 The Splitting Fields of a Simple Algebra.- 14.11 The Brauer Group. Factor Systems.- 15 General Ideal Theory of Commutative Rings.- 15.1 Noetherian Rings.- 15.2 Products and Quotients of Ideals.- 15.3 Prime Ideals and Primary Ideals.- 15.4 The General Decomposition Theorem.- 15.5 The First Uniqueness Theorem.- 15.6 Isolated Components and Symbolic Powers.- 15.7 Theory of Relatively Prime Ideals.- 15.8 Single-Primed Ideals.- 15.9 Quotient Rings.- 15.10 The Intersection ofall Powers of an Ideal.- 15.11 The Length of a Primary Ideal. Chains of Primary Ideals in Noetherian Rings.- 16 Theory of Polynomial Ideals.- 16.1 Algebraic Manifolds.- 16.2 The Universal Field.- 16.3 The Zeros of a Prime Ideal.- 16.4 The Dimension.- 16.5 Hilbert’s Nullstellensatz. Resultant Systems for Homogeneous Equations.- 16.6 Primary Ideals.- 16.7 Noether’s Theorem.- 16.8 Reduction of Multidimensional Ideals to Zero-Dimensional Ideals.- 17 Integral Algebraic Elements.- 17.1 Finite R-Modules.- 17.2 Integral Elements over a Ring.- 17.3 The Integral Elements of a Field.- 17.4 Axiomatic Foundation of Classical Ideal Theory.- 17.5 Converse and Extension of Results.- 17.6 Fractional Ideals.- 17.7 Ideal Theory of Arbitrary Integrally Closed Integral Domains.- 18 Fields with Valuations.- 18.1 Valuations.- 18.2 Complete Extensions.- 18.3 Valuations of the Field of Rational Numbers.- 18.4 Valuation of Algebraic Extension Fields: Complete Case.- 18.5 Valuation of Algebraic Extension Fields: General Case.- 18.6 Valuations of Algebraic Number Fields.- 18.7 Valuations of a Field ?(x) of Rational Functions.- 18.8 The Approximation Theorem.- 19 Algebraic Functions of One Variable.- 19.1 Series Expansions in the Uniformizing Variable.- 19.2 Divisors and Multiples.- 19.3 The Genus g.- 19.4 Vectors and Covectors.- 19.5 Differentials. The Theorem on the Speciality Index.- 19.6 The Riemann-Roch Theorem.- 19.7 Separable Generation of Function Fields.- 19.8 Differentials and Integrals in the Classical Case.- 19.9 Proof of the Residue Theorem.- 20 Topological Algebra.- 20.1 The Concept of a Topological Space.- 20.2 Neighborhood Bases.- 20.3 Continuity. Limits.- 20.4 Separation and Countability Axioms.- 20.5 Topological Groups.- 20.6 Neighborhoods of the Identity.- 20.7 Subgroupsand Factor Groups.- 20.8 T-Rings and Skew T-Fields.- 20.9 Group Completion by Means of Fundamental Sequences.- 20.10 Filters.- 20.11 Group Completion by Means of Cauchy Filters.- 20.12 Topological Vector Spaces.- 20.13 Ring Completion.- 20.14 Completion of Skew Fields.

Recenzii

From the reviews:
"This is the second volume of the English translation of B. L. van der Waerden’s classic textbook ‘Algebra’. … In fact, it represents the first softcover printing of the original translation which, on its part, had first appeared in1970. … this masterly exposition is still worth reading today, especially for beginners in commutative algebra and algebraic geometry. … It is van der Waerden’s inimitable style of presenting the principles of modern algebra that has survived all new fashions in algebra … ." (Werner Kleinert, Zentralblatt MATH, Vol. 1032 (7), 2004)
"In the glad to have you back department, I’m delighted that Springer has decided to reprint the two volumes of B.L.van der Waerden’s Algebra. Based in part on lectures by Emmy Noether and Emil Artin, this is the book that brought ‘abstract algebra’ to the mathematical world. … the book reflects the excitement that accompanied the birth of axiomatic algebra. … a book to treasure. I am glad it’s back." (MAA-Online, March, 2004)