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 TermsII. 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. TranspositionsIII. 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 CIV. 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 TheoremV. 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 DeterminantVI. 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 FractionsVII. Rings of Real and Complex Polynomials § 1. The Fundamental Theorem of Algebra 1. Introduction 2. The Fundamental Theorem of Algebra 3.