Introduction to Modern Algebra and Its Applications.

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.

The book provides an introduction to modern abstract algebra and its applications. It covers all major topics of classical theory of numbers, groups, rings, fields and finite dimensional algebras.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.