Includes index.

Machine generated contents note: 1.Preliminaries -- 1.1.Basic Ideas of Set Theory -- 1.2.Functions -- 1.3.Equivalence Relations and Partitions -- 1.4.A Note on Natural Numbers -- Review Exercises -- 2.Algebraic Structure of Numbers -- 2.1.The Set of Integers -- 2.2.Congruences of Integers -- 2.3.Rational Numbers -- Review Exercises -- 3.Basic Notions of Groups -- 3.1.Definitions and Examples -- 3.2.Basic Properties -- 3.3.Subgroups -- 3.4.Generating Sets -- Review Exercises -- 4.Cyclic Groups -- 4.1.Cyclic Groups -- 4.2.Subgroups of Cyclic Groups -- Review Exercises -- 5.Permutation Groups -- 5.1.Symmetric Groups -- 5.2.Dihedral Groups -- 5.3.Alternating Groups -- Review Exercises -- 6.Counting Theorems -- 6.1.Lagrange's Theorem -- 6.2.Conjugacy Classes of a Group -- Review Exercises -- 7.Group Homomorphisms -- 7.1.Examples and Basic Properties -- 7.2.Isomorphisms -- 7.3.Cayley's Theorem -- Review Exercises -- 8.The Quotient Group -- 8.1.Normal Subgroups -- 8.2.Quotient Groups -- Note continued: 8.3.Fundamental Theorem of Group Homomorphisms -- Review Exercises -- 9.Finite Abelian Groups -- 9.1.Direct Products of Groups -- 9.2.Cauchy's Theorem -- 9.3.Structure Theorem of Finite Abelian Groups -- Review Exercises -- 10.Group Actions -- 10.1.Definition and Basic Properties -- 10.2.Orbits and Stabilizers -- 10.3.Burnside's Formula -- Review Exercises -- 11.Sylow Theorems and Applications -- 11.1.The Three Sylow Theorems -- 11.2.Applications of Sylow Theorems -- Review Exercises -- 12.Introduction to Group Presentations -- 12.1.Free Groups and Free Abelian Groups -- 12.2.Generators and Relations -- 12.3.Classification of Finite Groups of Small Orders -- Review Exercises -- 13.Types of Rings -- 13.1.Definitions and Examples -- 13.2.Matrix Rings -- Review Exercises -- 14.Ideals and Quotient Rings -- 14.1.Ideals -- 14.2.Quotient Rings -- Review Exercises -- 15.Ring Homomorphisms -- 15.1.Ring Homomorphisms -- 15.2.Direct Products of Rings -- Note continued: 15.3.The Quotient Field of an Integral Domain -- Review Exercises -- 16.Polynomial Rings -- 16.1.Polynomial Rings in the Indeterminates -- 16.2.Properties of the Polynomial Rings of One Variable -- 16.3.Principal Ideal Domains and Euclidean Domains -- Review Exercises -- 17.Factorization -- 17.1.Irreducible and Prime Elements -- 17.2.Unique Factorization Domains -- 17.3.Polynomial Extensions of Factorial Domains -- Review Exercises -- 18.Introduction to Modules -- 18.1.Modules and Submodules -- 18.2.Linear Maps and Quotient Modules -- 18.3.Direct Sums of Modules -- Review Exercises -- 19.Free Modules -- 19.1.Free Modules -- 19.2.Determinant -- Review Exercises -- 20.Vector Spaces over Arbitrary Fields -- 20.1.A Brief Review on Vector Spaces -- 20.2.A Brief Review on Linear Transformations -- Review Exercises -- 21.Field Extensions -- 21.1.Algebraic or Transcendental? -- 21.2.Finite and Algebraic Extensions -- Note continued: 21.3.Construction with Straightedge and Compass -- Review Exercises -- 22.All About Roots -- 22.1.Zeros of Polynomials -- 22.2.Uniqueness of Splitting Fields -- 22.3.Algebraically Closed Fields -- 22.4.Multiplicity of Roots -- 22.5.Finite Fields -- Review Exercises -- 23.Galois Pairing -- 23.1.Galois Groups -- 23.2.The Fixed Subfields of a Galois Group -- 23.3.Fundamental Theorem of Galois Pairing -- Review Exercises -- 24.Applications of the Galois Pairing -- 24.1.Fields of Invariants -- 24.2.Solvable Groups -- 24.3.Insolvability of the Quintic -- Review Exercises.

Text in English