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