Algebra

PART 1: SET THEORY
Ch 0. Logical reasoning
1. The concept of logical perfection
2. The real language of mathematics
3. Elementary logical operations
4. Axioms and theorems
5. Logical axioms and tautologies
6. Substitution in a relation
7. Quantifiers
8. Rules for quantifiers
9. The Hilbert operation. Criteria of formation
Exercises on Ch 0
Ch 1. The relations of equality and membership
1. The relation of equality
2. The relation of membership
3. Subsets of a set
4. The empty set
5. Sets of one and two elements
6. The set of subsets of a given set
Exercises on Ch 1
Ch 2. The notion of a function
1. Ordered pairs
2. The Cartesian product of two sets
3. Graphs and functions
4. Direct and inverse images
5. Restrictions and extensions of functions
6. Composition of mappings
7. Injective mappings
8. Surjective and bijective mappings
9. Functions of several variables
Exercises on Ch 2
Ch 3. Unions and intersections
1. The union and intersection of two sets
2. The union of a family of sets
3. The intersection of a family of sets
Exercises on Ch 3
Ch 4. Equivalence relations
1. Equivalence relations
2. Quotient of a set by an equivalence relation
3. Functions defined on a quotient set
Exercises on Ch 4
Ch 5. Finite sets and integers
1. Equipotent sets
2. The cardinal of a set
3. Operations on cardinals
4. Finite sets and natural numbers
5. The set ℕ of the natural numbers
6. Mathematical induction
7. Combinatorial analysis
8. The rational integers
9. The rational numbers
Exercises on Ch 5
PART II: GROUPS, RINGS, FIELDS
Ch 6. Laws of composition
1. Laws of composition; associativity and commutativity
2. Reflexible elements
Ch 7. Groups
1. Definition of a group. Examples
2. Direct product of groups
3. Subgroups of a group
4. The intersection of subgroups. Generators
5. Permutations and transpositions
6. Cosets
7. The number of permutations of n objects
8. Homomorphisms
9. The kernel and image of a homomorphism
10. Application to cyclic groups
11. Groups operating on a set
Exercises on Ch 7
Ch 8. Rings and fields
1. Definition of a ring. Examples
2. Integral domains and fields
3. The ring of integers modulo p
4. The binomial theorem
5. Expansion of a product of sums
6. Ring homorphisms
Exercises on Ch 8
Ch 9. Complex numbers
1. Square roots
2. Preliminaries
3. The ring K[d^.5]
4. Units in a quadratic extension
5. The case of a field
6. Geometrical representation of complex numbers
1. Multiplication formulae for trigonometric functions
Exercises on Ch 9
PART III: MODULES OVER A RING
Ch 10. Modules and vector spaces
1. Definition of a module over a ring
2. Examples of modules
3. Submodules; vector subspaces
4. Right modules and left modules
Ch 11. Linear relations in a module
1. Linear combinations
2. Finitely generated modules
3. Linear relations
4. Free modules. Bases
5. Infinite linear combinations
Exercises on Ch 10 and 11
Ch 12. Linear mappings. Matrices
1. Homomorphisms
2. Homomorphisms of a finitely generated free module into an arbitrary module
3. Homomorphisms and matrices
4. Examples of homomorphisms and matrices
Ch 13. Addition of homomorphisms and matrices
1. The additive group Hom (L, M)
2. Addition of matrices
Ch 14. Products of matrices
1. The ring of endomorphisms of a module
2. The product of two matrices
3. Rings of matrices
4. Matrix notation for homomorphisms
Exercises on Ch 12, 13 and 14
Ch 15. Invertible matrices and change of basis
1. The group of automorphisms of a module
2. The groups GL(n, K)
3. Examples: the groups GL(1, K) and GL(2, K)
4. Change of basis. Transition matrices
5. Effect of change of bases on the matrix of a homomorphism
Exercises on Ch 15
Ch 16. The transpose of a linear mapping
1. The dual of a module
2. The dual of a finitely generated free module
3. The bidual of a module
4. The transpose of a homomorphism
5. The transpose of a matrix
Exercises on Ch 16
Ch 17. Sums of submodules
1. The sum of two submodules
2. Direct product of modules
3. Direct sum of submodules
4. Direct sums and projections
Exercises on Ch 17
PART IV: FINITE-DIMENSIONAL VECTOR SPACES
Ch 18. Finiteness theorems
1. Homomorphisms whose kernel and image are finitely generated
2. Finitely generated modules over a Noetherian ring
3. Submodules of a free module over a principal ideal domain
4. Applications to systems of linear equations
5. Other characterizations of Noetherian rings
Exercises on Ch 18
Ch 19. Dimension
1. Existence of bases
2. Definition of a vector subspace by means of linear equations
3. Conditions for consistency of a system of linear equations
4. Existence of linear relations
5. Dimension
6. Characterizations of bases and dimension
7. Dimensions of the kernel and image of a homomorphism
8. Rank of a homomorphism; rank of a family of vectors; rank of a matrix
9. Computation of the rank of a matrix
10. Calculation of the dimension of a vector subspace from its equations
Exercises on Ch 19
Ch 20. Linear equations
1. Notation and terminology
2. The rank of a system of linear equations. Conditions for the existence of solutions
3. The associated homogeneous system
4. Cramer systems
5. Systems of independent equations: reduction to a Cramer system
Exercises on Ch 20
PART V: DETERMINANTS
Ch 21. Multilinear functions
1. Definition of multilinear mappings
2. The tensor product of multilinear mappings
3. Some algebraic identities
4. The case of finitely generated free modules
5. The effect of a change of basis on the components of a tensor
Exercises on Ch 21
Ch 22. Alternating bilinear mappings
1. Alternating bilinear mappings
2. The case of finitely generated free modules
3. Alternating trilinear mappings
4. Expansion with respect to a basis
Exercises on Ch 22
Ch 23. Alternating multilinear mappings
1. The signature of a permutation
2. Antisymmetrization of a function of several variables
3. Alternating multilinear mappings
4. Alternating p-linear functions on a module isomorphic to K^p
5. Determinants
6. Characterization of bases of a finite-dimensional vector space
7. Alternating multilinear mappings: the general case
8. The criterion for linear independence
9. Conditions for consistency of a system of linear equations
Exercises on Ch 23
Ch 24. Determinants
1. Fundamental properties of determinants
2. Expansion of a determinant along a row or column
3. The adjugate matrix
4. Cramer's formulae
Exercises on Ch 24
Ch 25. Affine spaces
1. The vector space of translations
2. Affine spaces associated with a vector space
3. Barycentres in an affine space
4. Linear varieties in an affine space
5. Generation of a linear variety by means of lines
6. Finite-dimensional affine spaces. Affine bases
7. Calculation of the dimension of a linear variety
8. Equations of a linear variety in affine coordinates
PART VI: POLYNOMIALS AND ALGEBRAIC EQUATIONS
Ch 26. Algebraic relations
1. Monomials and polynomials in the elements of a ring
2. Algebraic relations
3. The case of fields
Exercises on Ch 26
Ch 27. Polynomial rings
1. Preliminaries on the case of one variable
2. Polynomials in one indeterminate
3. Polynomial notation
4. Polynomials in several indeterminates
5. Partial and total degrees
6. Polynomials with coefficients in an integral domain
Ch 28. Polynomial functions
1. The values of a polynomial
2. The sum and product of polynomial functions
3. The case of an infinite field
Exercises on Ch 27 and 28
Ch 29. Rational fractions
1. The field of fractions of an integral domain: preliminaries
2. Construction of the field of fractions
3. Verification of the field axioms
4. Embedding the ring K in its field of fractions
5. Rational fractions with coefficients in a field
6. Values of a rational fraction
Exercises on Ch 29
Ch 30. Derivations. Taylor's formula
1. Derivations of a ring
2. Derivations of a polynomial ring
3. Partial derivatives
4. Derivation of composite functions
5. Taylor's formula
6. The characteristic of a field
7. Multiplicities of the roots of an equation
Exercises on Ch 30
Ch 31. Principal ideal domains
1. Highest common factor
2. Coprime elements
3. Least common multiple
4. Existence of prime divisors
5. Properties of extremal elements
6. Uniqueness of the decomposition into prime factors
7. Calculation of h.c.f. and l.c.m. by means of prime factorization
8. Decomposition into partial fractions over a principal ideal domain
Exercises on Ch 31
Ch 32. Division of polynomials
1. Division of polynomials in one variable
2. Ideals in a polynomial ring in one indeterminate
3. The h.c.f. and l.c.m. of several polynomials. Irreducible polynomials
4. Application to rational fractions
Exercises on Ch 32
Ch 33. The roots of an algebraic equation
1. The maximum number of roots
2. Algebraically closed fields
3. Number of roots of an equation with coefficients in an algebraically closed field
4. Irreducible polynomials with coefficients in an algebraically closed field
5. Irreducible polynomials with real coefficients
6. Relations between the coefficients and the roots of an equation
Exercises on Ch 33
PART VII: REDUCTION OF MATRICES
Ch 34. Eigenvalues
1. Definition of eigenvectors and eigenvalues
2. The characteristic polynomial of a matrix
3. The form of the characteristic polynomial
4. The existence of eigenvalues
5. Reduction to triangular form
6. The case in which all the eigenvalues are simple
7. Characterization of diagonalizable endomorphisms
Exercises on Ch 34
Ch 35. The canonical form of a matrix
1. The Cayley-Hamilton theorem
2. Decomposition into nilpotent endomorphisms
3. The structure of nilpotent endomorphisms
4. Jordan's theorem
Exercises on Ch 35
Ch 36. Hermitian forms
1. Sesquilinear forms; hermitian forms
2. Non-degenerate forms
3. The adjoint of a homomorphism
4. Orthogonality with respect to a non-degenerate hermitian form
5. Orthogonal bases
6. Orthonormal bases
7. Automorphisms of a hermitian form
8. Automorphisms of a positive definite hermitian form. Reduction to diagonal form
9. Isotropic vectors and indefinite forms
10. The Cauchy-Schwarz inequality
Exercises on Ch 36
Bibliography
Index of Notation
Index of Terminology