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