Hjem

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.