Basic Modern Algebra with Applications by Mahima Ranjan Adhikari and Avishek Adhikari
MathSchoolinternational.com contain houndreds of
Free Math e-Books. Which cover almost all topics of mathematics. To see an extisive list of
Basic Algebra eBooks . We hope mathematician or person who’s interested in mathematics like these books.
Basic Modern Algebra with Applications by
Mahima Ranjan Adhikari , Institute for Mathematics, Bioinformatics, Information Technology And Computer Science (IMBIC), Kolkata, West Bengal, India and
Avishek Adhikari , Department of Pure Mathematics, University of Calcutta, Kolkata, West Bengal, India.
This book is designed to serve as a basic text of modern algebra at the undergraduate level. Modern mathematics facilitates unification of different areas of mathematics. It is characterized by its emphasis on the systematic study of a number of abstract mathematical structures. Modern algebra provides a language for almost all disciplines in contemporary mathematics.
Basic Modern Algebra with Applications by
Mahima Ranjan Adhikari and
Avishek Adhikari
cover the following topics.
1. Prerequisites: Basics of Set Theory and Integers
1.1 Sets: Introductory Concepts
1.2 Relations on Sets
1.2.1 Equivalence Relation
1.2.2 Partial Order Relations
1.2.3 Operations on Binary Relations
1.2.4 Functions or Mappings . 18
1.3 Countability and Cardinality of Sets
1.3.1 Continuum Hypothesis
1.3.2 Exercises
1.4 Integers
1.5 Congruences
1.5.1 Exercises
1.6 Additional Reading
References
2. Groups: Introductory Concepts
2.1 Binary Operations
2.2 Semigroups
2.2.1 Topological Semigroups
2.2.2 Fuzzy Ideals in a Semigroup
2.2.3 Exercises
2.3 Groups
2.4 Subgroups and Cyclic Groups
2.5 Lagrange’s Theorem
2.6 Normal Subgroups, Quotient Groups and Homomorphism
Theorems
2.7 Geometrical Applications
2.7.1 Symmetry Groups of Geometric Figures in Euclidean Plane
2.7.2 Group of Rotations of the Sphere
2.7.3 Clock Arithmetic
2.8 Free Abelian Groups and Structure Theorem
2.8.1 Exercises
2.9 Topological Groups, Lie Groups and Hopf Groups
2.9.1 Topological Groups
2.9.2 Lie Groups
2.9.3 Hops’s Groups or H-Groups
2.10 Fundamental Groups
2.10.1 A Generalization of Fundamental Groups
2.11 Exercises
2.12 Exercises (Objective Type)
2.13 Additional Reading
References
3. Actions of Groups, Topological Groups and Semigroups
3.1 Actions of Groups
3.2 Group Actions to Counting and Sylow’s Theorems
3.2.1 p-Groups and Cauchy’s Theorem
3.2.2 Class Equation and Sylow’s Theorems
3.2.3 Exercises
3.3 Actions of Topological Groups and Lie Groups
3.3.1 Exercises
3.4 Actions of Semigroups and State Machines
3.4.1 Exercises
3.5 Additional Reading
References
4. Rings: Introductory Concepts
4.1 Introductory Concepts
4.2 Subrings
4.3 Characteristic of a Ring
4.4 Embedding and Extension for Rings
4.5 Power Series Rings and Polynom
5. Ideals of Rings: Introductory Concepts
5.1 Ideals: Introductory concepts
5.2 QuotientRings
5.3 Prime Ideals and Maximal Ideals
5.4 Local Rings
5.5 Application to Algebraic Geometry
5.5.1 Affine Algebraic Sets
5.5.2 Ideal of a Set of Points in Kn
5.5.3 Affine Variety
5.5.4 The Zariski Topology in Kn
5.6 Chinese Remainder Theorem
5.7 Ideals of C(X)
5.8 Exercises
5.9 Additional Reading
References
6. Factorization in Integral Domains and in Polynomial Rings
6.1 Divisibility
6.2 Euclidean Domains
6.3 Factorization of Polynomials over a UFD
6.4 Supplementary Examples (SE-I)
6.5 Exercises
6.6 Additional Reading
References
7. Rings with Chain Conditions
7.1 Noetherian and Artinian Rings
7.2 An Application of Hilbert Basis Theorem to Algebraic Geometry
7.3 An Application of Cohen’s Theorem
7.4 Supplementary Examples (SE-I)
7.5 Exercises
7.6 Additional Reading
References
8. Vector Spaces
8.1 Introductory Concepts
8.2 Subspaces
8.3 Quotient Spaces
8.3.1 Geometrical Interpretation of Quotient Spaces
8.4 Linear Independence and Bases
8.4.1 Geometrical Interpretation
8.4.2 Coordinate System
8.4.3 Affine Set
8.4.4 Exercises
8.5 Linear Transformations and Associated Algebra
8.5.1 Algebra over a Field
8.6 Correspondence Between Linear Transformations and Matrices
8.7 Eigenvalues and Eigenvectors
8.8 The Cayley–Hamilton Theorem
8.9 Jordan Canonical Form
8.10 Inner Product Spaces
8.10.1 Geometry in Rn and Cn
8.10.2 Inner Product Spaces: Introductory Concepts
8.11 Hilbert Spaces
8.12 Quadratic Forms
8.12.1 Quadratic Forms: Introductory Concepts
8.13 Exercises
8.14 Additional Reading
References
9. Modules
9.1 Introductory Concepts
9.2 Submodules
9.3 Module Homomorphisms
9.4 Quotient Modules and Isomorphism Theorems
9.5 Modules of Homomorphisms
9.6 Free Modules, Modules over PID and Structure Theorems
9.6.1 Free Modules
9.6.2 Modules over PID
9.6.3 Structure Theorems
9.7 Exact Sequences
9.8 Modules with Chain Conditions
9.9 Representations
9.10 Worked-Out Exercises
9.10.1 Exercises
9.11 Homology and Cohomology Modules
9.11.1 Exercises
9.12 Topology on Spectrum of Modules and Rings
9.12.1 Spectrum of Modules
9.12.2 Spectrum of Rings and Associated Schemes
9.13 Additional Reading
References
10. Algebraic Aspects of Number Theory
10.1 A Brief History of Prime Numbers
10.2 Some Properties of Prime Numbers
10.2.1 Prime Number Theorem
10.2.2 Twin Primes
10.3 Multiplicative Functions
10.3.1 Euler phi-Function
10.3.2 Sum of Divisor Functions and Number of Divisor
Functions
10.4 Group of Units
10.5 Quadratic Residues and Quadratic Reciprocity
10.6 Fermat Numbers
10.7 Perfect Numbers and Mersenne Numbers
10.8 Analysis of the Complexity of Algorithms
10.8.1 Asymptotic Notation
10.8.2 Relational Properties Among the Asymptotic Notations
10.8.3 Poly-time and Exponential-Time Algorithms
10.8.4 Notations for Algorithms
10.8.5 Analysis of Few Important Number Theoretic Algorithms
10.8.6 Euclidean and Extended Euclidean Algorithms
10.8.7 Modular Arithmetic
10.8.8 Square and Multiply Algorithm
10.9 Primality Testing
10.9.1 Deterministic Primality Testing
10.9.2 AKS Algorithm
10.10 Probabilistic or Randomized Primality Testing
10.10.1 Solovay–Strassen Primality Testing
10.10.2 Pseudo-primes and Primality Testing Based on Fermat’s Little Theorem
10.10.3 Miller–Rabin Primality Testing
10.11 Exercise
10.12 Additional Reading
References
11. Algebraic Numbers
11.1 Field Extension
11.1.1 Exercises
11.2 Finite Fields
11.2.1 Exercises
11.3 Algebraic Numbers
11.4 Exercises
11.5 Algebraic Integers
11.6 Gaussian Integers
11.7 Algebraic Number Fields
11.8 Quadratic Fields
11.9 Exercises
11.10 Additional Reading
References
12. Introduction to Mathematical Cryptography
12.1 Introduction to Cryptography
12.2 Kerckhoffs’ Law
12.3 Cryptanalysis
12.3.1 Brute-Force Search
12.4 Some Classical Cryptographic Schemes
12.4.1 Caesarian Cipher or Shift Cipher
12.4.2 Affine Cipher
12.4.3 Substitution Cipher
12.4.4 Vigenère Cipher
12.4.5 Hill Cipher
12.5 Introduction to Public-Key Cryptography
12.5.1 Discrete Logarithm Problem (DLP)
12.5.2 Diffie–Hellman Key Exchange
12.6 RSA Public Key Cryptosystem
12.6.1 Algorithm for RSA Cryptosystem (Rivest et al. 1978)
12.6.2 Sketch of the Proof of the Decryption
12.6.3 Some Attacks on RSA Cryptosystem
12.7 ElGamal Cryptosystem
12.7.1 Algorithm for ElGamal Cryptosystem
12.8 Rabin Cryptosystem
12.8.1 Square Roots Modulo N
12.8.2 Algorithm for a variant of Rabin Cryptosystem
12.9 Digital Signature
12.9.1 RSA Based Digital Signature Scheme
12.10 Oblivious Transfer
12.10.1 OT Based on RSA
12.11 Secret Sharing
12.11.1 Shamir’s Threshold Secret Sharing Scheme
12.12 Visual Cryptography
12.12.1 (2, 2)-Visual Cryptographic Scheme
12.12.2 Visual Threshold Schemes
12.12.3 The Model for Black and White VCS
12.12.4 Basis Matrices
12.12.5 Share Distribution Algorithm
12.12.6 (2,n)-Threshold VCS
12.12.7 Applications of Linear Algebra to Visual Cryptographic Schemes for (n,n)-VCS
12.12.8 Applications of Linear Algebra to Visual Cryptographic Schemes for General Access Structure
12.13 Open-Source Software: SAGE
12.13.1 Sage Implementation of RSA Cryptosystem
12.13.2 SAGE Implementation of ElGamal Plulic Key Cryptosystem
12.13.3 SAGE Implementation of Rabin Plulic Key Cryptosystem
12.14 Exercises
12.15 Additional Reading
References
Appendix A Some Aspects of Semirings
A.1 Introductory Concepts
A.2 More Results on Semirings
A.3 Ring Congruences and Their Characterization
A.4 k-Regular Semirings
A.5 The Ring Completion of a Semiring
A.6 Structure Spaces of Semirings
A.7 Worked-Out Exercises and Exercises
A.8 Additional Reading
References
Appendix B Category Theory
B.1 Categories
B.2 Special Morphisms
B.3 Functors
B.4 Natural Transformations
B.5 Presheaf and Sheaf
B.6 Exercises
B.7 Additional Reading
References
Appendix C A Brief Historical Note
C.1 Additional Reading
References
Index
Start Now or
Open for Download
Download Similar Books
Math Books of BASIC ALGEBRA