EBOOK

Algebra


€ 36,99
 
kartoniert
Lieferbar innerhalb von 2-3 Tagen
Oktober 2003

Beschreibung

Beschreibung

This beautiful text transformed the graduate teaching of algebra in Europe and the United States. It clearly and succinctly formulated the conceptual and structural insights which Noether had expressed so forcefully and combined it with the elegance and understanding with which Artin had lectured. This second volume of the English translation of B.L. van der Waerden's text Algebra is the first softcover printing of the original translation.

Inhaltsverzeichnis

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 of all 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 Subgroups and 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.

Portrait

Bartel van der Waerden, geb. am 2.2.1903 in Amsterdam, ging 1924 ging als Student nach Göttingen und wurde dort mit Emmy Noether und der abstrakten Algebra bekannt. Sein Hauptinteresse galt damals vor allem der Begründung der algebraischen Geometrie mit Hilfe der neuen algebraischen Methoden. Als er im Jahre 1926 als junger Doktor mit einem Rockefeller-Stipendium nach Hamburg kam, hatte er Gelegenheit, eine didaktisch hervorragende Algebra-Vorlesung von Emil Artin zu hören. Die Ausarbeitung, die er von dieser Vorlesung machte, wurde zum Kern des vorliegenden Werkes. Es erschien zuerst 1930-31 unter dem Titel 'Moderne Algebra' in der Sammlung 'Grundlehren der mathematischen Wissenschaften'. In der Folge wurde das Werk in die englische, russische und chinesische Sprache übersetzt. Im Jahre 1928 wurde der Autor Professor an der Universität Groningen. Seit 1951 lebte und arbeitete er bis zu seiner Emeritierung in Zürich als Professor an der dortigen Universität.

Pressestimmen

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)
EAN: 9780387406251
ISBN: 0387406255
Untertitel: Volume II. 1st ed. 1991. 1st softcover printing 2003. Book. Sprache: Englisch.
Verlag: Springer
Erscheinungsdatum: Oktober 2003
Seitenanzahl: 300 Seiten
Übersetzer/Sprecher: Übersetzt von J. R. Schulenberger
Format: kartoniert
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