Hjem

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.