Introduction to Higher Algebra

I. Introduction

§ 1. Functions

1. Sets

2. Functions

3. Operations on Functions

§ 2. Mathematical Induction

1. Positive Integers and the Mathematical Induction

2. Sequences and Inductive Definitions

§ 3. Sums and Products of an Arbitrary Number of Terms

II. Some Combinatorial Problems

§ 1. Permutations

1. Permutations

2. Permutations with Repetitions

§ 2. k Permutations

1. k-Permutations without Repetitions

2. k-Permutations with Repetitions

§ 3. Combinations

1. Combinations

2. Properties of the Function (nk)

3. Combinations with Repetitions

§ 4. Newton's Multinomial Formula

1. Newton's Binomial Formula

2. Newton's Multinomial Formula

§ 5. Multiplication of Permutations

1. Definition

2. Factorization of a Permutation into Cycles

3. Transpositions

III. Complex Numbers

§ 1. Fields

1. Number Fields

2. General Fields

3. Isomorphism of Fields

4. Field of Real Numbers

5. Geometric Interpretation of Real Numbers

§ 2. Introductory Remarks on Complex Numbers

§ 3. Definition of Complex Numbers

§ 4. Properties of Complex Numbers

1. Geometric Interpretation of Complex Numbers

2. The Modulus and Conjugate Complex Numbers

3. Trigonometric Representation of Complex Numbers

§ 5. Roots of Complex Numbers

1. Square Roots of Complex Numbers

2. Roots of Higher Degree of Complex Numbers

3. Primitive Roots of Unity

4. Remarks on Fields Contained in C

IV. Determinants

§ 1. Definition of a Determinant

1. Introduction

2. Inversions

3. Applications of Inversions to the Theory of Permutations

4. Definition of a Matrix

5. Definition of a Determinant

§ 2. Laplace Expansion

1. Minors

2. Laplace Expansion

§ 3. Properties of Determinants

§ 4. Examples

1. Simple Examples

2. Example of a Cyclic Determinant

3. Vandermonde Determinant

4. Characteristic Polynomial

§ 5. Cramer's Formulae

§ 6. General Laplace Theorem

§ 7. Cauchy's Theorem and its Generalizations

1. Cauchy's Theorem

2. Cyclic Determinant

3. Generalization of Cauchy's Theorem

V. Vector Spaces and Linear Equations

§ 1. Vector Spaces

1. Definition

2. Linear Independence

3. Linear Subspaces

4. Basis and Dimension

§ 2. Rank of a Matrix

1. Simplest Properties

2. Investigation of the Rank of a Matrix by means of Minors

3. Independence of the Field

§ 3. Linear Equations

1. General Systems of Linear Equations

2. Homogeneous Equations

§ 4. Axiomatic Definition of the Determinant

VI. Polynomials in One Variable

§ 1. Operations on Polynomials

1. Polynomials

2. Differentiation of Polynomials

3. Taylor's and Maclaurin's Formulae

§ 2. The Arithmetic of the Ring K[x]

1. The Arithmetic of Integers

2. Division of Polynomials

3. The Greatest Common Divisor and the Least Common Multiple of Two Polynomials

4. Irreducible Polynomials

§ 3. Roots of a Polynomial

1. Multiple Roots

2. Divisibility of a Polynomial by Linear Factors

3. Elimination of Multiple Roots

§ 4. Interpolation Formulae

1. Lagrange's Formula

2. Newton's Interpolation Formula

3. Some Notions on Finite Differences

4. Arithmetic Progressions of Higher Orders

§ 5. Rational Functions

1. Definitions

2. Partial Fractions

VII. Rings of Real and Complex Polynomials

§ 1. The Fundamental Theorem of Algebra

1. Introduction

2. The Fundamental Theorem of Algebra

3. Consequences of the Fundamental Theorem

§ 2. Polynomials of the Ring R[x]

1. Fundamental Theorems

2. Sturm's Theorem

3. Descartes' and Harriot's Theorems

§ 3. Quadratic Equations in the Domain of Complex Numbers

1. Simplification of Equations by Substitution

2. Quadratic Equations

§ 4. Cubic Equations

1. Cardan's Formulae

2. Consequences of Cardan's Formulae

3. Examples

§ 5. Equations of the Fourth Degree

§ 6. Reciprocal Equations

VIII. Ring of Rational Polynomials Algebraic and Transcendental Numbers

§ 1. Reduction of Polynomials with Rational Coefficients to Polynomials with Integral Coefficients

1. Rational Roots of Polynomials with Rational Coefficients

2. Primitive Polynomials and Gauss' Theorem

§ 2. Polynomials Irreducible in the Field W

1. Eisenstein's Criterion

2. Kronecker's Method

§ 3. Algebraic Numbers

1. Introduction

2. Definitions

3. The Field of Algebraic Numbers

§ 4. Transcendental Numbers

1. Introduction

2. Liouville Transcendental Numbers

3. Cantor's Proof of Existence of Transcendental Numbers

IX. Polynomials in Several Variables and Symmetric Functions

§ 1. The Arithmetic of the Ring K[χ1, χ2, ..., χn]

1. Polynomials and Rational Functions

2. Irreducible Polynomials

3. The Uniqueness of the Factorization

4. Rational Functions in Several Variables

§ 2. Symmetric Polynomials

1. Definitions and Examples of Symmetric Polynomials

2. Lexicographic Order

3. The Fundamental Theorem of the Theory of Symmetric Polynomials

4. Examples of Applications of the Fundamental Theorem on Symmetric Functions

5. Newton's Formulae

6. Algebraic Independence of Elementary Symmetric Polynomials

7. Tschirnhausen's Transformation

X. The Theory of Elimination

§ 1. The Resultant

1. The Definition of the Resultant

2. Examples

§ 2. Systems of Two Equations in Two Unknowns

1. The Elimination of One Unknown

2. Examples

§ 3. Points of Intersection of Algebraic Curves

1. Properties of the Resultant

2. Bézout's Theorem

XI. Quadratic and Hermitian Forms

§ 1. Introduction

§ 2. Linear Transformations

1. Definitions

2. Composition of Transformations and Multiplication of Matrices

3. Some Properties of Composition of Transformations and Product of Matrices

4. Groups of Transformations

5. Addition of Matrices

6. Sylvester's Theorem on the Rank of a Product of Matrices

§ 3. Quadratic Forms

1. Notation

2. Linear Transformations of Quadratic Forms

3. Canonical Form

4. Real Quadratic Forms

5. Definite Forms

§ 4. Orthogonal Transformations of Quadratic Forms

1. Definition of Orthogonal Transformations

2. Characteristic Polynomial

3. Reduction to a Canonical Form by means of Orthogonal Transformations

§ 5. Hermitian Forms and Unitary Transformations

1. Hermitian Forms

2. Unitary Transformations

Appendix Some Properties of Matrices and Quadratic Forms

1. Cofactor-Matrix

2. The Case of a Singular Matrix

3. The Rank of the Matrix Formed by Minors

4. The Rank of a Symmetric Matrix

5. Sylvester's Identity. Kronecker's and Jacobi's Theorems on Quadratic Forms

6. Gram Matrices

7. Cayley's and Hamilton's Theorem

Index