Abstract Algebra

1. Advanced Topics
   1. Galois Theory
      1. Fundamental Theorem of Galois Theory
         1. Understanding Galois Extensions
         2. Correspondence between Subfields and Subgroups
         3. Fixed Field of a Group of Automorphisms
      2. Applications to Solvability of Polynomials
         1. Characterizing Solvable Groups
         2. Criteria for Solvability by Radicals
         3. Implications for Quintic Equations
      3. Connection with Field Theory
         1. Extensions and Automorphism Groups
         2. Radical Extensions
         3. Composite and Separable Extensions
   2. Representation Theory
      1. Group Representations
         1. Definition and Examples
         2. Linear Representations and Matrices
         3. Schur's Lemma and Complete Reducibility
      2. Characters and Class Functions
         1. Character Tables
         2. Orthogonality Relations
         3. Induction and Restriction of Characters
      3. Representations of Specific Groups
         1. Symmetric Groups
         2. Cyclic Groups
         3. Representation of Abelian Groups
      4. Applications of Representation Theory
         1. In Quantum Mechanics
         2. Use in Chemistry for Molecular Symmetries
   3. Homological Algebra
      1. Chain Complexes
         1. Definition and Construction
         2. Homology Groups
         3. Short and Long Exact Sequences
      2. Exact Sequences
         1. Split Exact Sequences
         2. Applications in Algebra and Topology
         3. Snake Lemma and Excision
      3. Ext and Tor Functors
         1. Definitions and Properties
         2. Applications in Homology
         3. Extensions of Modules and Derived Functors
      4. Applications to Algebraic Topology
         1. Singular Homology
         2. Cohomology Theories
         3. Connection to K-theory
   4. Category Theory in Algebra
      1. Categories and Functors
         1. Definitions and Examples
         2. Functors between Categories
         3. Natural Transformations
      2. Limits and Colimits
         1. Constructing Limits and Colimits
         2. Universal Properties
         3. Examples in Various Categories
      3. Adjunctions and Monads
         1. Definitions and Applications
         2. Link with Universal Algebra
         3. Use in programming languages and type theory
      4. Applications in Algebra
         1. Reformulation of Algebraic Structures
         2. Use in Homological Algebra
         3. Interpretations in Set Theory and Logic