Introduction to Modern Algebra and Its Applications 1st Edition by Nadiya Gubareni ISBN 9780367820916 0367820919

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 Congmences

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 Grobner 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