Algebra I is a comprehensive textbook written by Alexey L. Gorodentsev. This book provides a thorough introduction to the subject of algebra, one of the most fundamental branches of mathematics.
The author has taken a clear and straightforward approach to teach algebra, with a focus on providing students with a solid foundation in the basic concepts and techniques. Throughout the book, the author explains the material in a step-by-step manner, with numerous examples and exercises to help students reinforce their understanding.
The book covers all the essential topics in algebra, including linear equations, quadratic equations, polynomials, and functions. Each topic is presented in a clear and concise manner, with an emphasis on helping students develop their problem-solving skills. The author also includes numerous real-world applications of algebra, demonstrating the importance of the subject and how it can be applied in various fields.
TABLE OF CONTENTS
1 Set-Theoretic and Combinatorial Background
- Sets and Maps
- Sets
- Maps
- Fibers of Maps
- Equivalence Classes
- Equivalence Relations
- Implicitly Defined Equivalences
- Compositions of Maps
- Composition Versus Multiplication
- Right Inverse Map and the Axiom of Choice
- Invertible Maps
- Transformation Groups
- Posets
- Partial Order Relations
- Well-Ordered Sets
- Zorn’s Lemma
2 Integers and Residues
- Fields, Rings, and Abelian Groups
- Definition of a Field
- Commutative Rings
- Abelian Groups
- Subtraction and Division
- The Ring of Integers
- Divisibility
- The Equation ax C by D k and the Greatest
- The Euclidean Algorithm
- Coprime Elements
- Rings of Residues
- Residue Classes Modulo n
- Zero Divisors and Nilpotents
- Invertible Elements in Residue Rings
- Residue Fields
- Direct Products of Commutative Groups and Rings
- Homomorphisms
- Homomorphisms of Abelian Groups
- Kernel of a Homomorphism
- Group of Homomorphisms
- Homomorphisms of Commutative Rings
- Homomorphisms of Fields
- Chinese Remainder Theorem
- Characteristic
- Prime Subfield
- Frobenius Endomorphism
.
3 Polynomials and Simple Field Extensions
- Formal Power Series
- Rings of Formal Power Series
- Algebraic Operations on Power Series
- Polynomials
- Differential Calculus
- Polynomial Rings
- Division
- Coprime Polynomials
- Euclidean Algorithm
- Roots of Polynomials
- Common Roots
- Multiple Roots
- Separable Polynomials
- Adjunction of Roots
- Residue Class Rings
- Algebraic Elements
- Algebraic Closure
- The Field of Complex Numbers
- The Complex Plane
- Complex Conjugation
- Trigonometry
- Roots of Unity and Cyclotomic Polynomials
- The Gaussian Integers
- Finite Fields
- Finite Multiplicative Subgroups in Fields
- Description of All Finite Fields
- Quadratic Residues
- Problems for Independent Solution to Chap. 3
4 Elementary Functions and Power Series Expansions
- Rings of Fractions
- Localization
- Field of Fractions of an Integral Domain
- Field of Rational Functions
- Simplified Fractions
- Partial Fraction Expansion
- Power Series Expansions of Rational Functions
- Linear Recurrence Relations
- Logarithm and Exponential
- The Logarithm
- The Exponential
- Power Function and Binomial Formula
- Todd’s Series and Bernoulli Numbers
- Action of Qd=dt on QŒt
- Bernoulli Numbers
- Fractional Power Series
- Puiseux Series
- Newton’s Method
5 Ideals, Quotient Rings, and Factorization
- Ideals
- Definition and Examples
- Noetherian Rings
- Quotient Rings
- Factorization Homomorphism
- Maximal Ideals and Evaluation Maps
- Prime Ideals and Ring Homomorphisms to Fields
- Finitely Generated Commutative Algebras
- Principal Ideal Domains
- Euclidean Domains
- Greatest Common Divisor
- Coprime Elements
- Irreducible Elements
- Unique Factorization Domains
- Irreducible Factorization
- Prime Elements
- GCD in Unique Factorization Domains
- Polynomials over Unique Factorization Domains
- Factorization of Polynomials with Rational Coefficients
- Reduction of Coefficients
- Kronecker’s Algorithm
6 Vectors
- Vector Spaces and Modules
- Definitions and Examples
- Linear Maps
- Proportional Vectors
- Bases and Dimension
- Linear Combinations
- Linear Dependence
- Basis of a Vector Space
- Infinite-Dimensional Vector Spaces
- Space of Linear Maps
- Kernel and Image
- Matrix of a Linear Map
- Vector Subspaces
- Codimension
- Linear Spans
- Sum of Subspaces
- Tranversal Subspaces
- Direct Sums and Direct Products
- Affine Spaces
- Definition and Examples
- Affinization and Vectorization
- Center of Mass
- Affine Subspaces
- Affine Maps
- Affine Groups
- Quotient Spaces
- Quotient by a Subspace
- Quotient Groups of Abelian Groups
7 Duality
- Dual Spaces
- Covectors
- Canonical Inclusion V ,! V
- Dual Bases
- Pairings
- Annihilators
- Dual Linear Maps
- Pullback of Linear Forms
- Rank of a Matrix
8 Matrices
- Associative Algebras over a Field
- Definition of Associative Algebra
- Invertible Elements
- Algebraic and Transcendental Elements
- Matrix Algebras
- Multiplication of Matrices
- Invertible Matrices
- Transition Matrices
- Gaussian Elimination
- Elimination by Row Operations
- Location of a Subspace with Respect to a Basis
- Gaussian Method for Inverting Matrices
- Matrices over Noncommutative Rings
9 Determinants
- Volume Forms
- Volume of an n-Dimensional Parallelepiped
- Skew-Symmetric Multilinear Forms
- Digression on Parities of Permutations
- Determinants
- Basic Properties of Determinants
- Determinant of a Linear Endomorphism
- Grassmannian Polynomials
- Polynomials in Skew-Commuting Variables
- Linear Change of Grassmannian Variables
- Laplace Relations
- Adjunct Matrix
- Row and Column Cofactor Expansions
- Matrix Inversion
- Cayley–Hamilton Identity
- Cramer’s Rules
10 Euclidean Spaces
- Inner Product
- Euclidean Structure
- Length of a Vector
- Orthogonality
- Gramians
- Gram Matrices
- Euclidean Volume
- Orientation
- Cauchy–Bunyakovsky–Schwarz Inequality
- Duality
- Isomorphism V ⥲ V Provided by Euclidean Structure
- Orthogonal Complement and Orthogonal Projection
- Metric Geometry
- Euclidean Metric
- Angles
- Orthogonal Group
- Euclidean Isometries
- Orthogonal Matrices
11 Projective Spaces
- Projectivization
- Points and Charts
- Global Homogeneous Coordinates
- Local Affine Coordinates
- Polynomials Revisited
- Polynomial Functions on a Vector Space
- Symmetric Algebra of a Vector Space
- Polynomial Functions on an Affine Space
- Affine Algebraic Varieties
- Projective Algebraic Varieties
- Homogeneous Equations
- Projective Closure of an Affine Hypersurface
- Space of Hypersurfaces
- Linear Systems of Hypersurfaces
- Complementary Subspaces and Projections
- Linear Projective Isomorphisms
- Action of a Linear Isomorphism on Projective Space
- Linear Projective Group
- Cross Ratio
- Action of the Permutation Group S4
- Special Quadruples of Points
- Harmonic Pairs of Points
12 Groups
- Definition and First Examples
- Cycles
- Cyclic Subgroups
- Cyclic Groups
- Cyclic Type of Permutation
- Groups of Figures
- Homomorphisms of Groups
- Group Actions
- Definitions and Terminology
- Orbits and Stabilizers
- Enumeration of Orbits
- Factorization of Groups
- Cosets
- Normal Subgroups
- Quotient Groups
13 Descriptions of Groups
- Generators and Relations
- Free Groups
- Presentation of a Group by Generators and Relators
- Presentations for the Dihedral Groups
- Presentations of the Groups of Platonic Solids
- Presentation of the Symmetric Group
- Complete Group of a Regular Simplex
- Bruhat Order
- Simple Groups and Composition Series
- Jordan–Hölder Series
- Finite Simple Groups
- Semidirect Products
- Semidirect Product of Subgroups
- Semidirect Product of Groups
- p-Groups and Sylow’s Theorems
- p-Groups in Action
- Sylow Subgroups
14 Modules over a Principal Ideal Domain
- Modules over Commutative Rings Revisited
- Free Modules
- Generators and Relations
- Linear Maps
- Matrices of Linear Maps
- Torsion
- Quotient of a Module by an Ideal
- Direct Sum Decompositions
- Semisimplicity
- Invariant Factors
- Submodules of Finitely Generated Free Modules
- Deduction of the Invariant Factors Theorem from the Smith Normal Form Theorem
- Uniqueness of the Smith Normal Form
- Gaussian Elimination over a Principal Ideal Domain
- Elementary Divisors
- Elementary Divisors Versus Invariant Factors
- Existence of the Canonical Decomposition
- Splitting Off Torsion
- Splitting Off p-Torsion
- Invariance of p-Torsion Exponents
- Description of Finitely Generated Abelian Groups
- Canonical Form of a Finitely Generated Abelian Group
- Abelian Groups Presented by Generators and Relations
15 Linear Operators
- Classification of Operators
- Spaces with Operators
- Invariant Subspaces and Decomposability
- Space with Operator as a kŒt-Module
- Elementary Divisors
- Minimal Polynomial
- Characteristic Polynomial
- Operators of Special Types
- Nilpotent Operators
- Semisimple Operators
- Cyclic Vectors and Cyclic Operators
- Eigenvectors and Eigenvalues
- Eigenspaces
- Diagonalizable Operators
- Annihilating Polynomials
- Jordan Decomposition
- Jordan Normal Form
- Root Decomposition
- Commuting Operators
- Nilpotent and Diagonalizable Components
- Functions of Operators
- Evaluation of Functions on an Operator
- Interpolating Polynomial
- Comparison with Analytic Approaches
16 Bilinear Forms
- Bilinear Forms and Correlations
- Space with Bilinear Form
- Gramians
- Left Correlation
- Nondegeneracy
- Kernels
- Nonsymmetric and (Skew)-Symmetric Forms
- Characteristic Polynomial and Characteristic Values
- Nondegenerate Forms
- Dual Bases
- Isotropic Subspaces
- Isometry Group
- Correspondence Between Forms and Operators
- Canonical Operator
- Adjoint Operators
- Reflexive Operators
- Orthogonals and Orthogonal Projections
- Orthogonal Projections
- Biorthogonal Direct Sums
- Classification of Nondegenerate Forms
- Symmetric and Skew-Symmetric Forms
- Orthogonals and Kernel
- Orthogonal Projections
- Adjoint Operators
- Form–Operator Correspondence
- Symplectic Spaces
- Symplectic Group
- Lagrangian Subspaces
- Pfaffian
17 Quadratic Forms and Quadrics
- Quadratic Forms and Their Polarizations
- Space with a Quadratic Form
- Gramian and Gram Determinant
- Kernel and Rank
- Sums of Squares
- Isotropic and Anisotropic Subspaces
- Hyperbolic Forms
- Orthogonal Geometry of Nonsingular Forms
- Isometries
- Reflections
- Quadratic Forms over Real and Simple Finite Fields
- Quadratic Forms over Fp
- Real Quadratic Forms
- How to Find the Signature of a Real Form
- Projective Quadrics
- Geometric Properties of Projective Quadrics
- Smooth Quadrics
- Polarities
- Affine Quadrics
- Projective Enhancement of Affine Quadrics
- Smooth Central Quadrics
- Paraboloids
- Simple Cones
- Cylinders
18 Real Versus Complex
- Realification
- Realification of a Complex Vector Space
- Comparison of Linear Groups
- Complexification
- Complexification of a Real Vector Space
- Complex Conjugation
- Complexification of Linear Maps
- Complex Eigenvectors
- Complexification of the Dual Space
- Complexification of a Bilinear Form
- Real Structures
- Complex Structures
- Hermitian Enhancement of Euclidean Structure
- Hermitian Structure
- Kähler Triples
- Completing Kähler Triples for a Given Euclidean Structure
- Hermitian Enhancement of Symplectic Structure
- Completing Kähler Triples for a Given
19 Hermitian Spaces
- Hermitian Geometry
- Gramians
- Gram–Schmidt Orthogonalization Procedure
- Cauchy–Schwarz Inequality
- Unitary Group
- Hermitian Volume
- Hermitian Correlation
- Orthogonal Projections
- Angle Between Two Lines
- Adjoint Linear Maps
- Hermitian Adjunction
- Euclidean Adjunction
- Normal Operators
- Orthogonal Diagonalization
- Normal Operators in Euclidean Space
- Polar and Singular Value Decompositions
- Polar Decomposition
- Exponential Cover of the Unitary Group
- Singular Value Decomposition
20 Quaternions and Spinors
- Complex 2 2 Matrices and Quaternions
- Mat2.C/ as the Complexification of Euclidean R4
- Algebra of Quaternions
- Real and Pure Imaginary Quaternions
- Quaternionic Norm
- Division
- Geometry of Quaternions
- Universal Covering
- Topological Comment
- Two Pencils of Hermitian Structures
- Spinors
- Geometry of Hermitian Enhancements of Euclidean R4
- Explicit Formulas
- Hopf Bundle
Download Now Book in PDF
One of the standout features of this book is its emphasis on practice. The author has included a wealth of problems and exercises, ranging from simple computational exercises to more challenging, conceptual problems. These problems are designed to help students develop their critical thinking and problem-solving skills and to deepen their understanding of algebra.
In addition to its clear and accessible approach, Algebra I is also highly accessible. The book is available in PDF format for students to download, making it easy for them to access and study the material. This also makes it a cost-effective option for students who are looking to save money on their textbook costs.
In conclusion, Algebra I is an excellent resource for students who are looking to learn about algebra. With its clear explanations, emphasis on practice, and real-world applications, this book is sure to help students develop a deep and lasting understanding of this important subject.
Note: This book is available in PDF format for students to download, but the chapters are not elaborately discussed in the article.
