Algebra  v. 1 ~ v. 2

B.L. van der Waerden ; based in part on lectures by E. Artin and E. Noether ; translated by Fred Blum and John R. Schulenberger

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.

「Nielsen BookData」より

This beautiful and eloquent 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 text is a reprinted version of the original English translation of the first volume of B.L. van der Waerden's Algebra.

「Nielsen BookData」より

[目次]

  • 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.

「Nielsen BookData」より

[目次]

  • 1 Numbers and Sets.- 1.1 Sets.- 1.2 Mappings. Cardinality.- 1.3 The Number Sequence.- 1.4 Finite and Countable (Denumerable) Sets.- 1.5 Partitions.- 2 Groups.- 2.1 The Concept of a Group.- 2.2 Subgroups.- 2.3 Complexes. Cosets.- 2.4 Isomorphisms and Automorphisms.- 2.5 Homomorphisms, Normal Subgroups, and Factor Groups.- 3 Rings and Fields.- 3.1 Rings.- 3.2 Homomorphism and Isomorphism.- 3.3 The Concept of a Field of Quotients.- 3.4 Polynomial Rings.- 3.5 Ideals. Residue Class Rings.- 3.6 Divisibility. Prime Ideals.- 3.7 Euclidean Rings and Principal Ideal Rings.- 3.8 Factorization.- 4 Vector Spaces and Tensor Spaces.- 4.1 Vector Spaces.- 4.2 Dimensional Invariance.- 4.3 The Dual Vector Space.- 4.4 Linear Equations in a Skew Field.- 4.5 Linear Transformations.- 4.6 Tensors.- 4.7 Antisymmetric Multilinear Forms and Determinants.- 4.8 Tensor Products, Contraction, and Trace.- 5 Polynomials.- 5.1 Differentiation.- 5.2 The Zeros of a Polynomial.- 5.3 Interpolation Formulae.- 5.4 Factorization.- 5.5 Irreducibility Criteria.- 5.6 Factorization in a Finite Number of Steps.- 5.7 Symmetric Functions.- 5.8 The Resultant of Two Polynomials.- 5.9 The Resultant as a Symmetric Function of the Roots.- 5.10 Partial Fraction Decomposition.- 6 Theory of Fields.- 6.1 Subfields. Prime Fields.- 6.2 Adjunction.- 6.3 Simple Field Extensions.- 6.4 Finite Field Extensions.- 6.5 Algebraic Field Extensions.- 6.6 Roots of Unity.- 6.7 Galois Fields (Finite Commutative Fields).- 6.8 Separable and Inseparable Extensions.- 6.9 Perfect and Imperfect Fields.- 6.10 Simplicity of Algebraic Extensions. Theorem on the Primitive Element.- 6.11 Norms and Traces.- 7 Continuation of Group Theory.- 7.1 Groups with Operators.- 7.2 Operator Isomorphisms and Operator Homomorphisms.- 7.1 The Two Laws of Isomorphism.- 7.4 Normal Series and Composition Series.- 7.5 Groups of Order pn.- 7.6 Direct Products.- 7.7 Group Characters.- 7.8 Simplicity of the Alternating Group.- 7.9 Transitivity and Primitivity.- 8 The Galois Theory.- 8.1 The Galois Group.- 8.2 The Fundamental Theorem of the Galois Theory.- 8.3 Conjugate Groups, Conjugate Fields, and Elements.- 8.4 Cyclotomic Fields.- 8.5 Cyclic Fields and Pure Equations.- 8.6 Solution of Equations by Radicals.- 8.7 The General Equation of Degree n.- 8.8 Equations of the Second, Third, and Fourth Degrees.- 8.9 Constructions with Ruler and Compass.- 8.10 Calculation of the Galois Group. Equations with a Symmetric Group.- 8.11 Normal Bases.- 9 Ordering and Well Ordering of Sets.- 9.1 Ordered Sets.- 9.2 The Axiom of Choice and Zorn's Lemma.- 9.3 The Well-Ordering Theorem.- 9.4 Transfinite Induction.- 10 Infinite Field Extensions.- 10.1 Algebraically Closed Fields.- 10.2 Simple Transcendental Extensions.- 10.3 Algebraic Dependence and Independence.- 10.4 The Degree of Transcendency.- 10.5 Differentiation of Algebraic Functions.- 11 Real Fields.- 11.1 Ordered Fields.- 11.2 Definition of the Real Numbers.- 11.3 Zeros of Real Functions.- 11.4 The Field of Complex Numbers.- 11.5 Algebraic Theory of Real Fields.- 11.6 Existence Theorems for Formally Real Fields.- 11.7 Sums of Squares.

「Nielsen BookData」より

この本の情報

書名 Algebra
著作者等 Artin, Emil
Noether, Emmy
Schulenberger, J. R.
Waerden, B. L. van der
Waerden B. L. van der
Blum Fred
書名別名 Moderne Algebra
巻冊次 v. 1
v. 2
出版元 Springer-Verlag
刊行年月 2003, c1991
版表示 1st ed. 1991. 2nd printing 2003
ページ数 2 v.
大きさ 24 cm
ISBN 0387406255
0387406247
NCID BA69963578
※クリックでCiNii Booksを表示
言語 英語
原文言語 ドイツ語
出版国 アメリカ合衆国
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