دانلود کتاب Introduction to Modern Algebra and Its Applications

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Cover Title Page Copyright Page Preface Contents 1. Elements of Number Theory 1.1 Divisibility of Integers. Division with Remainder 1.2 The Greatest Common Divisor. The Euclidean Algorithm 1.3 The Extended Euclidean Algorithm 1.4 Relatively Primes 1.5 Linear Diophantine Equations 1.6 Congruences and their Properties 1.7 Linear Congruences 1.8 Exercises References 2. Elements of Group Theory 2.1 Semigroups, Monoids and Groups 2.2 Subgroups. Cyclic Groups 2.3 Permutation Groups 2.4 Cosets. Lagrange's Theorem 2.5 Normal Subgroups and Quotient Groups 2.6 Group Homomorphisms 2.7 The Isomorphism Theorems 2.8 Exercises References 3. Examples of Groups 3.1 Cycle Notation and Cycle Decomposition of Permutations 3.2 Inversions, Parity and Order of a Permutation 3.3 Alternating Group 3.4 Cyclic Groups 3.5 Groups of Symmetries. The Dihedral Groups 3.6 Direct Product of Groups 3.7 Finite Abelian Groups 3.8 Exercises References 4. Elements of Ring Theory 4.1 Rings and Subrings 4.2 Integral Domains and Fields 4.3 Ideals and Ring Homomorphisms 4.4 Quotient Rings 4.5 Maximal Ideals. Prime Ideals 4.6 Principal Ideal Rings 4.7 Euclidean Domains. Euclidean Algorithm 4.8 Unique Factorization Domains 4.9 Chinese Remainder Theorem 4.10 Exercises References 5. Polynomial Rings in One Variable 5.1 Basic Definitions and Properties 5.2 Division with Remainder 5.3 Greatest Common Divisor of Polynomials 5.4 Factorization of Polynomials. Irreducible Polynomials 5.5 Roots of Polynomials 5.6 Polynomials over Rational Numbers 5.7 Quotient Rings of Polynomial Rings 5.8 Exercises References 6. Elements of Field Theory 6.1 A Field of Fractions of an Integral Domain 6.2 The Characteristic of a Field 6.3 Field Extensions 6.4 Algebraic Elements. Algebraic Extensions 6.5 Splitting Fields 6.6 Algebraically Closed Fields 6.7 Polynomials over Complex Numbers and Real Numbers 6.8 Exercises References 7. Examples of Applications 7.1 Euler's φ-function and its Properties 7.2 Euler's Theorem. Fermat's Little Theorem. Wilson's Theorem 7.3 Solving Linear Congruences by Euler's Method 7.4 Solving Systems of Linear Congruences 7.5 Lagrange's Interpolation Polynomials 7.6 Secret Sharing 7.7 Cryptographic Algorithm RSA 7.8 Exercises References 8. Polynomials in Several Variables 8.1 Polynomial Rings in Several Variables 8.2 Symmetric Polynomials 8.3 Noetherian Rings. Hilbert Basis Theorem 8.4 Monomial Order 8.5 Division Algorithm for Polynomials 8.6 Initial Ideals. Gröbner Basis 8.7 S-polynomials 8.8 Buchberger's Algorithm 8.9 Minimal and Reduced Gröbner Basis 8.10 Applications of Gröbner Bases 8.11 Exercises References 9. Finite Fields and their Applications 9.1 Properties of Finite Fields 9.2 Multiplicative Group of a Finite Field 9.3 Primitive Roots and Indexes. Discrete Logarithm Problem 9.4 Diffie-Hellman Scheme. ElGamal Cryptosystem 9.5 Error Detecting and Error Correcting Codes 9.6 Exercises References 10. Finite Dimensional Algebras 10.1 Quaternions and their Properties 10.2 Octonions-Cayley's Octaves 10.3 Algebras and their Properties 10.4 Division Algebras. Algebras with Involution. Composition Algebras 10.5 Cayley-Dickson Construction 10.6 Dual Numbers and Double Numbers 10.7 Clifford Algebras. Grassmann Algebras 10.8 Exercises References 11. Applications of Quaternions and Octonions 11.1 Square Sum Identities 11.2 Gaussian Integers 11.3 Fermat's Theorem on Sums of Two Squares 11.4 Lagrange's Four-Square Theorem 11.5 Trigonometric Form of Quaternions 11.6 Rotations and Quaternions 11.7 Exercises References Appendix A.1 Basic Concepts of Set Theory. Relations on Sets A.2 Operations on Sets. Algebraic Structures A.3 Vector Spaces Index