Learn abstract algebra concepts with clear lessons, examples, and quizzes
191
Sub Topics
217
MCQs
80
MCOs
158
True/False
86
Fill Blanks
29
Rearrange
79
Matching
37
Comprehensions
80
Flashcard Decks
Curriculum
What You'll Learn
01 Fundamentals and Set Theory 2 topics
1 Set Theory Basics
- Sets, Subsets, and Operations
- Functions and Relations
- Equivalence Relations and Partitions
- Partially Ordered Sets
2 Mathematical Logic and Proof Techniques
- Direct Proof
- Proof by Contradiction
- Induction and Recursive Definitions
- The Principle of Well-Ordering
02 Group Theory 10 topics
1 Introduction to Groups
- Definition and Examples
- Elementary Properties
- Symmetry Groups
- Dihedral Groups
2 Subgroups
- Definition and Examples
- Cyclic Subgroups
- Centralizers and Normalizers
- Center of a Group
3 Cyclic Groups
- Properties of Cyclic Groups
- Subgroups of Cyclic Groups
- Classification of Cyclic Groups
4 Permutation Groups
- Symmetric Groups
- Cycles and Transpositions
- Parity of Permutations
- Alternating Groups
5 Cosets and Lagrange's Theorem
- Left and Right Cosets
- Index of a Subgroup
- Lagrange's Theorem and Applications
- Fermat's Little Theorem
6 Normal Subgroups and Quotient Groups
- Normal Subgroups
- Quotient Groups
- Homomorphism Theorems
- Simple Groups
7 Group Homomorphisms
- Definition and Properties
- Kernel and Image
- Isomorphism Theorems
- Automorphisms
8 Group Actions
- Definition and Examples
- Orbits and Stabilizers
- Burnside's Lemma
- Applications to Counting Problems
9 The Sylow Theorems
- p-Groups and Cauchy's Theorem
- Sylow p-Subgroups
- Sylow's Theorems
- Applications to Classification
10 Direct and Semidirect Products
- Direct Products
- Internal and External Direct Products
- Semidirect Products
- Group Extensions
03 Ring Theory 8 topics
1 Introduction to Rings
- Definition and Examples
- Elementary Properties
- Units and Zero Divisors
- Fields and Division Rings
2 Subrings and Ideals
- Subrings
- Ideals
- Principal Ideals
- Prime and Maximal Ideals
3 Ring Homomorphisms
- Definition and Properties
- Kernel and Image
- Isomorphism Theorems
- Quotient Rings
4 Polynomial Rings
- Polynomials over a Ring
- Division Algorithm
- Irreducibility
- Unique Factorization Domains
5 Euclidean Domains
- Definition and Properties
- Euclidean Algorithm
- Greatest Common Divisors
- Bezout's Identity
6 Principal Ideal Domains
- Properties of PIDs
- Relation to Euclidean Domains
- Factorization in PIDs
- Applications
7 Unique Factorization Domains
- Prime and Irreducible Elements
- Properties of UFDs
- Gauss's Lemma
- Eisenstein's Criterion
8 Modules over Rings
- Definition and Examples
- Submodules and Quotient Modules
- Module Homomorphisms
- Free Modules
04 Field Theory 6 topics
1 Introduction to Fields
- Definition and Examples
- Subfields
- Prime Fields
- Characteristic of a Field
2 Field Extensions
- Algebraic and Transcendental Extensions
- Degree of an Extension
- Algebraic Closure
- Separable and Inseparable Extensions
3 Splitting Fields
- Definition and Construction
- Existence and Uniqueness
- Normal Extensions
- Splitting Fields of Separable Polynomials
4 Finite Fields
- Construction of Finite Fields
- Properties of Finite Fields
- Subfields of Finite Fields
- Applications to Coding Theory
5 Galois Theory
- Galois Groups
- The Fundamental Theorem of Galois Theory
- Solvability by Radicals
- Applications to Classical Problems
6 Cyclotomic Extensions
- Cyclotomic Polynomials
- Cyclotomic Fields
- Gauss Sums
- Applications to Number Theory
05 Advanced Group Theory 4 topics
1 Group Representations
- Definition and Basic Properties
- Character Theory
- Irreducible Representations
- Applications
2 Classification of Finite Groups
- Solvable and Nilpotent Groups
- p-Groups
- Classification of Groups of Small Order
- Simple Groups
3 Free Groups
- Definition and Construction
- Group Presentations
- Word Problems
- Applications to Combinatorial Group Theory
4 Infinite Groups
- Torsion and Torsion-Free Groups
- Locally Finite Groups
- Growth of Groups
- Geometric Group Theory
06 Advanced Ring and Module Theory 3 topics
1 Noetherian and Artinian Rings
- Ascending and Descending Chain Conditions
- Properties of Noetherian Rings
- Properties of Artinian Rings
- Hilbert's Basis Theorem
2 Dedekind Domains
- Definition and Properties
- Fractional Ideals
- Ideal Class Group
- Applications to Algebraic Number Theory
3 Advanced Module Theory
- Projective Modules
- Injective Modules
- Flat Modules
- Tensor Products
07 Commutative Algebra 4 topics
1 Localization
- Localization of Rings and Modules
- Properties of Localization
- Local Rings
- Applications
2 Primary Decomposition
- Primary Ideals
- Primary Decomposition Theorem
- Uniqueness of Primary Decomposition
- Applications
3 Integral Extensions
- Integral Elements
- Integral Closure
- Going-Up and Going-Down Theorems
- Normalization
4 Dimension Theory
- Krull Dimension
- Height of Prime Ideals
- Dimension of Polynomial Rings
- Regular Local Rings
08 Homological Algebra 3 topics
1 Categories and Functors
- Categories
- Functors
- Natural Transformations
- Adjoint Functors
2 Exact Sequences
- Definition and Properties
- Short Exact Sequences
- Long Exact Sequences
- The Snake Lemma
3 Derived Functors
- Ext and Tor
- Cohomology of Groups
- Spectral Sequences
- Applications
09 Applications of Abstract Algebra 4 topics
1 Algebraic Geometry
- Affine and Projective Varieties
- Schemes
- Sheaves and Cohomology
- Intersection Theory
2 Algebraic Number Theory
- Number Fields
- Algebraic Integers
- Class Number and Units
- Diophantine Equations
3 Cryptography
- Public Key Cryptosystems
- Elliptic Curve Cryptography
- Lattice-Based Cryptography
- Post-Quantum Cryptography
4 Coding Theory
- Linear Codes
- Cyclic Codes
- Reed-Solomon Codes
- LDPC Codes and Turbo Codes
10 Special Topics in Abstract Algebra 4 topics
1 Lie Algebras
- Definition and Examples
- Structure Theory
- Root Systems
- Classification of Simple Lie Algebras
2 Algebraic K-Theory
- K0 and K1
- Higher K-Groups
- Milnor K-Theory
- Applications
3 Non-commutative Algebra
- Division Algebras
- Group Algebras
- Quantum Groups
- Hopf Algebras
4 Model Theory and Universal Algebra
- First-Order Logic
- Models and Theories
- Varieties and Equational Classes
- Ultraproducts and Compactness
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Abstract Algebra
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