Algebra II is a crucial subject in mathematics that builds upon the concepts and skills learned in Algebra I. It covers a wide range of topics, including polynomials, rational functions, exponential and logarithmic functions, matrices, and conic sections, to name a few.

Alexey L. Gorodentsev’s book, Algebra II, is a comprehensive guide to these algebraic concepts. It is written in a clear and concise manner, making it easy for students to understand the concepts and techniques presented in each chapter. The book is designed for high school students who are taking Algebra II, as well as for those who are preparing for the advanced placement (AP) exams in mathematics.

The book begins by reviewing the basic concepts of Algebra I, including linear and quadratic equations, systems of linear equations, and graphing. It then goes on to cover the more advanced topics of Algebra II, such as polynomials, rational functions, and exponential and logarithmic functions. Each chapter includes numerous examples and exercises, as well as explanations of key concepts, to help students understand and practice the techniques presented.

**TABLE OF CONTENTS**

**Tensor Products**

- Multilinear Maps
- Multilinear Maps Between Free Modules
- Universal Multilinear Map
- Tensor Product of Modules
- Existence of Tensor Product
- Linear Maps as Tensors
- Tensor Products of Abelian Groups
- Commutativity, Associativity, and Distributivity Isomorphisms
- Tensor Product of Linear Maps
- Tensor Product of Modules Presented by Generators and Relations

**2 Tensor Algebras**

- Free Associative Algebra of a Vector Space
- Contractions
- Complete Contraction
- Partial Contractions
- Linear Support and Rank of a Tensor
- Quotient Algebras of a Tensor Algebra
- Symmetric Algebra of a Vector Space
- Symmetric Multilinear Maps
- The Exterior Algebra of a Vector Space
- Alternating Multilinear Maps
- Symmetric and Alternating Tensors
- Symmetrization and Alternation
- Standard Bases
- Polarization of Polynomials
- Evaluation of Polynomials on Vectors
- Combinatorial Formula for Complete Polarization
- Duality
- Derivative of a Polynomial Along a Vector
- Polars and Tangents of Projective Hypersurfaces
- Linear Support of a Homogeneous Polynomial
- Polarization of Grassmannian Polynomials
- Duality
- Partial Derivatives in an Exterior Algebra
- Linear Support of a Homogeneous Grassmannian Polynomial
- Grassmannian Varieties and the Plücker Embedding
- The Grassmannian as an Orbit Space

**3 Symmetric Functions**

- Symmetric and Sign Alternating Polynomials
- Elementary Symmetric Polynomials
- Complete Symmetric Polynomials
- Newton’s Sums of Powers
- Generating Function for the pk
- Transition from ek and hk to pk
- Giambelli’s Formula
- Pieri’s Formula
- The Ring of Symmetric Functions

**4 Calculus of Arrays, Tableaux, and Diagrams**

- Arrays
- Notation and Terminology
- Vertical Operations
- Commutation Lemma
- Condensing
- Condensed Arrays
- Bidense Arrays and Young Diagrams
- Young Tableaux
- Yamanouchi Words
- Fiber Product Theorem
- Action of the Symmetric Group on DU-Sets
- DU-Sets and DU-Orbits
- Action of Sm D Aut.J/
- Combinatorial Schur Polynomials
- The Littlewood–Richardson Rule
- The Jacobi–Trudi Identity
- Transition from e and h to s
- The Inner Product on ƒ

**5 Basic Notions of Representation Theory**

- Representations of a Set of Operators
- Associative Envelope
- Decomposability and (Semi)/Simplicity
- Homomorphisms of Representations
- Representations of Associative Algebras
- Double Centralizer Theorem
- Digression: Modules Over Noncommutative Rings
- Isotypic Components
- Representations of Groups
- Direct Sums and Tensor Constructions
- Representations of Finite Abelian Groups
- Reynolds Operator
- Group Algebras
- Center of a Group Algebra
- Isotypic Decomposition of a Finite Group Algebra
- Schur Representations of General Linear Groups
- Action of GL.V/ Sn on V ̋n
- The Schur–Weyl Correspondence

**6 Representations of Finite Groups in Greater Detail**

- Orthogonal Decomposition of a Group Algebra
- Invariant Scalar Product and Plancherel’s Formula
- Irreducible Idempotents
- Characters
- Definition, Properties, and Examples of Computation
- The Fourier Transform
- Ring of Representations
- Induced and Coinduced Representations
- Restricted and Induced Modules Over Associative Algebras
- Induced Representations of Groups
- The Structure of Induced Representations
- Coinduced Representations

**7 Representations of Symmetric Groups**

- The action of Sn on Filled Young Diagrams
- Row and Column Subgroups Associated with a Filling
- Young Symmetrizers sT D rT cT
- Young Symmetrizes
- Modules of Tabloids
- Specht Modules
- Description and Irreducibility
- Standard Basis Numbered by Young Tableaux
- Representation Ring of Symmetric Groups
- Littlewood–Richardson Product
- Scalar Product on <
- The Isometric Isomorphism < ⥲ ƒ
- Dimensions of Irreducible Representations
- sl2-Modules
- Lie Algebras
- Universal Enveloping Algebra
- Representations of Lie Algebras
- Finite-Dimensional Simple sl2-Modules
- Semisimplicity of Finite-Dimensional sl2-Modules

**9 Categories and Functors**

- Objects and Morphisms
- Mono-, Epi-, and Isomorphisms
- Reversing of Arrows
- Functors
- Covariant Functors
- Presheaves
- The Functors Hom
- Natural Transformations
- Equivalence of Categories
- Representable Functors
- Definitions via Universal Properties
- Adjoint Functors
- Tensor Products Versus Hom Functors
- Limits of Diagrams
- (Co) completeness
- Filtered Diagrams
- Functorial Properties of (Co) limits
- Extensions of Commutative Rings
- Integral Elements
- Definition and Properties of Integral Elements
- Algebraic Integers
- Normal Rings
- Applications to Representation Theory
- Algebraic Elements in Algebras
- Transcendence Generators

**11 Affine Algebraic Geometry**

- Systems of Polynomial Equations
- Affine Algebraic–Geometric Dictionary
- Coordinate Algebra
- Maximal Spectrum
- Pullback Homomorphisms
- Zariski Topology
- Irreducible Components
- Rational Functions
- The Structure Sheaf
- Principal Open Sets as Affine Algebraic Varieties
- Geometric Properties of Algebra Homomorphisms
- Closed Immersions
- Dominant Morphisms
- Finite Morphisms
- Normal Varieties

**12 Algebraic Manifolds**

- Definitions and Examples
- Structure Sheaf and Regular Morphisms
- Closed Submanifolds
- Families of Manifolds
- Separated Manifolds
- Rational Maps
- Projective Varieties
- Resultant Systems
- 1 Resultant of Two Binary Forms
- Closeness of Projective Morphisms
- Finite Projections
- Dimension of an Algebraic Manifold
- Dimensions of Subvarieties
- Dimensions of Fibers of Regular Maps
- Dimensions of Projective Varieties

**13 Algebraic Field Extensions**

- Finite Extensions
- Primitive Extensions
- Separability
- Extensions of Homomorphisms
- Splitting Fields and Algebraic Closures
- Normal Extensions
- Compositum
- Automorphisms of Fields and the Galois Correspondence

**14 Examples of Galois Groups**

- Straightedge and Compass Constructions
- Effect of Accessory Irrationalities
- Galois Groups of Polynomials
- Galois Resolution
- Reduction of Coefficients
- Galois Groups of Cyclotomic Fields
- Frobenius Elements
- Cyclic Extensions
- Solvable Extensions
- Generic Polynomial of Degree n
- Solvability of Particular Polynomials

**Download Now Book in PDF**

One of the strengths of Gorodentsev’s book is its clear and concise writing style. He explains complex concepts in a way that is easy to understand, using real-world examples and practical applications. This makes it a great resource for students who struggle with mathematics, as well as for those who are looking to get a deeper understanding of the subject.

Another great feature of this book is its comprehensive coverage of Algebra II concepts. Whether you are a student who needs to learn the basics or an advanced learner looking for a comprehensive review, this book has everything you need. It provides a strong foundation for students who want to move on to more advanced mathematical subjects, such as Calculus or Linear Algebra.

We are pleased to announce that Algebra II by Alexey L. Gorodentsev is available in PDF format, so students can easily download and access it. Whether you prefer to study on your own or with a teacher, this book is a valuable resource for anyone who wants to gain a deeper understanding of Algebra II.

In conclusion, Algebra II by Alexey L. Gorodentsev is a comprehensive guide to the subject of Algebra II. Its clear writing style, comprehensive coverage of concepts, and numerous examples and exercises make it an ideal resource for students, teachers, and anyone looking to improve their understanding of this important subject.