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Basic Modern Algebra with Applications by Mahima Ranjan Adhikari and Avishek Adhikari



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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

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