Abstract Algebra

1. Related Mathematical Concepts
   1. Linear Algebra
      1. Vector Spaces and Linear Maps
         1. Definition of Vector Spaces
            1. Scalars and field properties
            2. Examples in various dimensions
         2. Basis and Dimension
            1. Basis vectors and linear independence
            2. Dimension theorem and applications
         3. Linear Maps
            1. Definition and properties
            2. Kernel and image
            3. Rank-Nullity Theorem
         4. Change of Basis
            1. Transition matrices
            2. Effect on vector representation
      2. Matrices and Determinants
         1. Definition and Types of Matrices
            1. Square, diagonal, symmetric, and identity matrices
            2. Block matrices and applications
         2. Matrix Operations
            1. Addition, multiplication, and scalar multiplication
            2. Transpose and its properties
         3. Inverse of a Matrix
            1. Conditions for invertibility
            2. Calculation methods: Gauss-Jordan elimination
         4. Determinants
            1. Properties and calculation methods
            2. Cofactor expansion and Laplace's formula
            3. Applications of determinants: volume, linear dependence
         5. Eigenvalues and Eigenvectors
            1. Characteristic polynomial
            2. Diagonalization of matrices
            3. Applications in differential equations and dynamical systems
   2. Number Theory
      1. Algebraic Number Fields
         1. Definition and examples: quadratic fields
         2. Ring of integers
         3. Units and the unit group
         4. Factorization
            1. Primes in number fields
            2. Unique factorization and its failure
      2. Diophantine Equations
         1. Linear Diophantine equations
            1. Solutions and properties
            2. GCD method
         2. Quadratic Diophantine equations
            1. Pell's equation and solutions
            2. Applications in continued fractions
         3. Higher-degree Diophantine equations
            1. Fermat's Last Theorem: historical and modern perspectives
            2. Elliptic curves and rational solutions
      3. Congruences and Modular Arithmetic
         1. Basic properties and theorems
         2. Chinese Remainder Theorem
         3. Applications in cryptography
   3. Combinatorics
      1. Algebraic Combinatorics
         1. Enumerative combinatorics
            1. Counting techniques: permutations, combinations
            2. Binomial coefficients and Pascal's triangle
         2. Graph theory
            1. Graphs and their properties
            2. Coloring problems and applications
         3. Matroid theory
            1. Structures and properties
            2. Applications to optimization problems
      2. Generating Functions
         1. Ordinary generating functions
            1. Definition and manipulations
            2. Applications in counting problems
         2. Exponential generating functions
            1. Use in solving recurrence relations
            2. Applications in differential equations
         3. Applications in finding closed forms for series
            1. Technique for summation and convergence
            2. Analytic combinatorics and asymptotic behavior