Abstract Algebra Group Theory, Rings & Fields | BMLabs | BMLabs Mathematics

Subject knowledge map

Learning structure

Explore every published chapter, section, and lesson in this subject through one keyboard-accessible visual hierarchy.

Interactive concept atlas

Abstract Algebra Learning Map

240 concepts · 239 relationships · auto mode

Concept map ready to load

The graph engine loads only when this learning map approaches the viewport.

Accessible text outline

Concepts

Abstract Algebrasubject
Sets, Relations, and Integerschapter
Sets and Set Operationssection
Algebra of Integers and Divisibilitysection
Relations and Their Propertiessection
Partially Ordered Sets and Latticessection
Functions and Types of Functionssection
Binary Operations and Algebraic Structuressection
Introduction to Groupschapter
Definition and Examples of Groupssection
Additive Groups Number Systemssubsection
Binary Operations on Setssubsection
First Nonexamples of Groupssubsection
Four Group Axiomssubsection
Group Definitionsubsection
Multiplicative Groups Nonzero Numberssubsection
Verify Group Axiomssubsection
Standard Examples of Groupssection
Additive Groups Modulo Nsubsection
Alternating Groupssubsection
Groups From Number Systemssubsection
Klein Four Groupsubsection
Permutation Groupssubsection
Roots of Unity Groupssubsection
Semigroup Conditions for Groupssection
Finite Cancellative Semigroupssubsection
Idempotents in Finite Semigroupssubsection
Semigroup Equation Criterion for Groupssubsection
Semigroupssubsection
Unique Idempotent Semigroupssubsection
Elementary Properties of Groupssection
Cancellation Laws in Groupssubsection
Equations in Groupssubsection
Idempotent Elements in Groupssubsection
Identity Element in Groupssubsection
Inverse Element in Groupssubsection
Properties of Inversessubsection
Abelian Groupssection
Abelian Groupsubsection
Elements of Order Twosubsection
Groups of Exponent Twosubsection
Inverse Product Criterionsubsection
Power Criteria for Abelian Groupssubsection
Cyclic Groupssection
Cyclic Groupsubsection
Cyclic Groups Are Abeliansubsection
Even Cyclic Groupssubsection
Finite and Infinite Cyclic Groupssubsection
Generators of Finite Cyclic Groupssubsection
Prime Order Cyclic Groupssubsection
Order of an Elementsection
Finite and Infinite Ordersubsection
Order of Inverse Elementsubsection
Order of Powerssubsection
Orders of Products and Conjugatessubsection
Powers Equal to Identitysubsection
Prime Order Elementssubsection
Powers of Elements in Groupssection
Conjugation and Powerssubsection
Integral Power Lawssubsection
Laws of Powers in Groupssubsection
Positive and Negative Powerssubsection
Power Equations in Groupssubsection
Permutation Groupschapter
Permutations and Cycle Notationsection
Symmetric and Alternating Groupssection
Subgroups and Normal Subgroupschapter
Subgroupssection
Finite Subgroupssubsection
Intersections of Subgroupssubsection
Products of Subgroupssubsection
Subgroup Criteriasubsection
Subgroupssubsection
Unions of Subgroupssubsection
Special Subgroupssection
Centraliser of an Elementsubsection
Centre and Centraliserssubsection
Centre of a Groupsubsection
Cosets of Subgroupssection
Cardinality and Correspondence of Cosetssubsection
Coset Membership and Equality Criteriasubsection
Cosets and Intersections of Subgroupssubsection
Cosets as Equivalence Classessubsection
Cosets as Partitionssubsection
Cosets Determined by Elements of a Subgroupsubsection
Equal and Disjoint Cosetssubsection
Left and Right Cosetssubsection
Index and Lagrange Theoremsection
Divisors and Subgroup Orderssubsection
Groups of Prime Ordersubsection
Index of a Subgroupsubsection
Lagrange Theoremsubsection
Order of an Element in a Finite Groupsubsection
Powers in a Finite Groupsubsection
Applications of Lagrange Theoremsection
Index and Intersections of Subgroupssubsection
Index Multiplication Formulasubsection
Product Formula for Finite Subgroupssubsection
Relatively Prime Subgroup Orderssubsection
Subgroups of Prime Indexsubsection
Subgroups of Prime Power Order Groupssubsection
Double Cosetssection
Cardinality of Double Cosetssubsection
Double Coset Definitionsubsection
Double Cosets as Equivalence Classessubsection
Normal Subgroupssection
Central and Cyclic Sources of Normal Subgroupssubsection
Definition and First Examples of Normal Subgroupssubsection
Index Two and Coset Criteria for Normalitysubsection
Normalisers of Subgroupssubsection
Normality Inside Related Subgroupssubsection
Products and Intersections of Normal Subgroupssubsection
Quotient Groupssection
Abelian and Cyclic Quotient Groupssubsection
Abelian Quotients and Commutatorssubsection
Construction of Quotient Groupssubsection
Normal Subgroups in Quotient Groupssubsection
Order of a Quotient Groupsubsection
Quotients by Central Subgroupssubsection
Homomorphisms and Isomorphisms of Groupschapter
Homomorphisms of Groupssection
Homomorphism of Groupssubsection
Basic Properties of Homomorphismssubsection
Kernel of a Homomorphismsubsection
Epimorphism, Monomorphism, Isomorphism, and Automorphismsubsection
Natural Homomorphismsubsection
Isomorphism Tests and Examplessubsection
Classification of Cyclic Groupssubsection
Homomorphisms Between Cyclic Groupssubsection
Power Maps on Groupssubsection
Cayley Theorem and Permutation Groupssubsection
Isomorphism and Correspondence Theoremssection
Factorization Through Quotientssubsection
First Isomorphism Theoremsubsection
Homomorphic Images and Normal Subgroupssubsection
Second Isomorphism Theoremsubsection
Third Isomorphism Theoremsubsection
Correspondence Theoremsubsection
Group Actionssection
Group Actions and G-Setssubsection
Orbits and Stabilizerssubsection
Orbit-Stabilizer Formula and Orbit Countingsubsection
Permutation Representations and Extended Cayley Theoremsubsection
Fixed Points and Burnside's Lemmasubsection
Direct Product of Groupschapter
External Direct Productsection
Internal Direct Productsection
Sylow Theoremschapter
Conjugacy Classessection
Centralizers and Conjugacy of Elementssubsection
Class Equation and Conjugacy Class Decompositionsubsection
Conjugate Subgroups and Invariancesubsection
Normalizers and Counting Conjugate Subgroupssubsection
Cauchy's Theorem and p Groupssection
Cauchy's Theoremsubsection
Subgroups of Finite Abelian Groupssubsection
p-Groups and p-Subgroupssubsection
Centers of Finite p-Groupssubsection
Groups of Order p Squared and p Cubedsubsection
Subgroups of Finite p-Groupssubsection
Normal Subgroups from Cauchy's Theoremsubsection
Computing p-Subgroupssubsection
Sylow Theoremssection
Sylow's First Theoremsubsection
Sylow p-Subgroupssubsection
Conjugates and Normality of Sylow p-Subgroupssubsection
Sylow's Second Theoremsubsection
Sylow's Third Theoremsubsection
Sylow Subgroups in Normal Subgroups and Quotientssubsection
Applications of Sylow Countingsubsection
Direct Products of Sylow Subgroupssubsection
Some Applications of the Sylow Theoremssection
Simple Groups and Non-Simplicity Testssubsection
Non-Simplicity of Groups of Small Ordersubsection
Simple Groups of Order 60 and A5subsection
Groups of Order pqsubsection
Dihedral Groups and Groups of Order 2psubsection
Groups of Small Ordersubsection
Cyclicity from Sylow Subgroupssubsection
Central Sylow Subgroups and Normality Applicationssubsection
True-or-False Applications of Sylow Theoremssubsection
Solvable and Nilpotent Groupschapter
Solvable Groupssection
Nilpotent Groupssection
Finitely Generated Abelian Groupschapter
Finite Abelian Groupssection
Structure Theorem for Finitely Generated Abelian Groupssection
Introduction to Ringschapter
Definition and Basic Properties of Ringssection
Important Classes of Ringssection
Subrings, Ideals, and Homomorphismschapter
Subrings and Subfieldssection
Ideals and Quotient Ringssection
Ring Homomorphisms and Isomorphismssection
Ring Embeddingschapter
Embedding of Ringssection
Direct Sum of Ringschapter
Direct Sum and Product of Ringssection
Polynomial Ringschapter
Polynomial Rings over Fields and Ringssection
Euclidean Domainschapter
Euclidean Domains and Division Algorithmsection
Greatest Common Divisors in Domainssection
Prime and Irreducible Elementssection
Unique Factorization Domainschapter
Unique Factorization Domainssection
Factorization of Polynomials over UFDsection
Irreducibility Criteria for Polynomialssection
Maximal, Prime, and Primary Idealschapter
Maximal, Prime, and Primary Idealssection
Jacobson Semisimple Ringssection
Noetherian and Artinian Ringschapter
Noetherian and Artinian Ringssection
Modules and Vector Spaceschapter
Modules and Vector Spacessection
Rings of Matriceschapter
Full Matrix Ringssection
Triangular Matrix Ringssection
Field Extensionschapter
Field Extensions and Degreesection
Splitting Fieldssection
Algebraically Closed Fieldssection
Multiplicity of Rootschapter
Multiplicity of Rootssection
Finite Fieldschapter
Finite Fieldssection
Galois Theory and Applicationschapter
Normal Extensionssection
Galois Theorysection
Roots of Unity and Cyclotomic Polynomialssection
Solvability of Polynomials by Radicalssection
Geometric Constructionschapter
Geometric Constructionssection
Coding Theorychapter
Binary Codessection
Polynomial and Cyclic Codessection
Bose-Chaudhuri-Hocquenghem Codessection
Gröbner Baseschapter
Affine Varietiessection
Gröbner Basessection

Relationships

Abstract Algebra --contains--> Sets, Relations, and Integers
Sets, Relations, and Integers --contains--> Sets and Set Operations
Sets, Relations, and Integers --contains--> Algebra of Integers and Divisibility
Sets, Relations, and Integers --contains--> Relations and Their Properties
Sets, Relations, and Integers --contains--> Partially Ordered Sets and Lattices
Sets, Relations, and Integers --contains--> Functions and Types of Functions
Sets, Relations, and Integers --contains--> Binary Operations and Algebraic Structures
Abstract Algebra --contains--> Introduction to Groups
Introduction to Groups --contains--> Definition and Examples of Groups
Definition and Examples of Groups --contains--> Additive Groups Number Systems
Definition and Examples of Groups --contains--> Binary Operations on Sets
Definition and Examples of Groups --contains--> First Nonexamples of Groups
Definition and Examples of Groups --contains--> Four Group Axioms
Definition and Examples of Groups --contains--> Group Definition
Definition and Examples of Groups --contains--> Multiplicative Groups Nonzero Numbers
Definition and Examples of Groups --contains--> Verify Group Axioms
Introduction to Groups --contains--> Standard Examples of Groups
Standard Examples of Groups --contains--> Additive Groups Modulo N
Standard Examples of Groups --contains--> Alternating Groups
Standard Examples of Groups --contains--> Groups From Number Systems
Standard Examples of Groups --contains--> Klein Four Group
Standard Examples of Groups --contains--> Permutation Groups
Standard Examples of Groups --contains--> Roots of Unity Groups
Introduction to Groups --contains--> Semigroup Conditions for Groups
Semigroup Conditions for Groups --contains--> Finite Cancellative Semigroups
Semigroup Conditions for Groups --contains--> Idempotents in Finite Semigroups
Semigroup Conditions for Groups --contains--> Semigroup Equation Criterion for Groups
Semigroup Conditions for Groups --contains--> Semigroups
Semigroup Conditions for Groups --contains--> Unique Idempotent Semigroups
Introduction to Groups --contains--> Elementary Properties of Groups
Elementary Properties of Groups --contains--> Cancellation Laws in Groups
Elementary Properties of Groups --contains--> Equations in Groups
Elementary Properties of Groups --contains--> Idempotent Elements in Groups
Elementary Properties of Groups --contains--> Identity Element in Groups
Elementary Properties of Groups --contains--> Inverse Element in Groups
Elementary Properties of Groups --contains--> Properties of Inverses
Introduction to Groups --contains--> Abelian Groups
Abelian Groups --contains--> Abelian Group
Abelian Groups --contains--> Elements of Order Two
Abelian Groups --contains--> Groups of Exponent Two
Abelian Groups --contains--> Inverse Product Criterion
Abelian Groups --contains--> Power Criteria for Abelian Groups
Introduction to Groups --contains--> Cyclic Groups
Cyclic Groups --contains--> Cyclic Group
Cyclic Groups --contains--> Cyclic Groups Are Abelian
Cyclic Groups --contains--> Even Cyclic Groups
Cyclic Groups --contains--> Finite and Infinite Cyclic Groups
Cyclic Groups --contains--> Generators of Finite Cyclic Groups
Cyclic Groups --contains--> Prime Order Cyclic Groups
Introduction to Groups --contains--> Order of an Element
Order of an Element --contains--> Finite and Infinite Order
Order of an Element --contains--> Order of Inverse Element
Order of an Element --contains--> Order of Powers
Order of an Element --contains--> Orders of Products and Conjugates
Order of an Element --contains--> Powers Equal to Identity
Order of an Element --contains--> Prime Order Elements
Introduction to Groups --contains--> Powers of Elements in Groups
Powers of Elements in Groups --contains--> Conjugation and Powers
Powers of Elements in Groups --contains--> Integral Power Laws
Powers of Elements in Groups --contains--> Laws of Powers in Groups
Powers of Elements in Groups --contains--> Positive and Negative Powers
Powers of Elements in Groups --contains--> Power Equations in Groups
Abstract Algebra --contains--> Permutation Groups
Permutation Groups --contains--> Permutations and Cycle Notation
Permutation Groups --contains--> Symmetric and Alternating Groups
Abstract Algebra --contains--> Subgroups and Normal Subgroups
Subgroups and Normal Subgroups --contains--> Subgroups
Subgroups --contains--> Finite Subgroups
Subgroups --contains--> Intersections of Subgroups
Subgroups --contains--> Products of Subgroups
Subgroups --contains--> Subgroup Criteria
Subgroups --contains--> Subgroups
Subgroups --contains--> Unions of Subgroups
Subgroups and Normal Subgroups --contains--> Special Subgroups
Special Subgroups --contains--> Centraliser of an Element
Special Subgroups --contains--> Centre and Centralisers
Special Subgroups --contains--> Centre of a Group
Subgroups and Normal Subgroups --contains--> Cosets of Subgroups
Cosets of Subgroups --contains--> Cardinality and Correspondence of Cosets
Cosets of Subgroups --contains--> Coset Membership and Equality Criteria
Cosets of Subgroups --contains--> Cosets and Intersections of Subgroups
Cosets of Subgroups --contains--> Cosets as Equivalence Classes
Cosets of Subgroups --contains--> Cosets as Partitions
Cosets of Subgroups --contains--> Cosets Determined by Elements of a Subgroup
Cosets of Subgroups --contains--> Equal and Disjoint Cosets
Cosets of Subgroups --contains--> Left and Right Cosets
Subgroups and Normal Subgroups --contains--> Index and Lagrange Theorem
Index and Lagrange Theorem --contains--> Divisors and Subgroup Orders
Index and Lagrange Theorem --contains--> Groups of Prime Order
Index and Lagrange Theorem --contains--> Index of a Subgroup
Index and Lagrange Theorem --contains--> Lagrange Theorem
Index and Lagrange Theorem --contains--> Order of an Element in a Finite Group
Index and Lagrange Theorem --contains--> Powers in a Finite Group
Subgroups and Normal Subgroups --contains--> Applications of Lagrange Theorem
Applications of Lagrange Theorem --contains--> Index and Intersections of Subgroups
Applications of Lagrange Theorem --contains--> Index Multiplication Formula
Applications of Lagrange Theorem --contains--> Product Formula for Finite Subgroups
Applications of Lagrange Theorem --contains--> Relatively Prime Subgroup Orders
Applications of Lagrange Theorem --contains--> Subgroups of Prime Index
Applications of Lagrange Theorem --contains--> Subgroups of Prime Power Order Groups
Subgroups and Normal Subgroups --contains--> Double Cosets
Double Cosets --contains--> Cardinality of Double Cosets
Double Cosets --contains--> Double Coset Definition
Double Cosets --contains--> Double Cosets as Equivalence Classes
Subgroups and Normal Subgroups --contains--> Normal Subgroups
Normal Subgroups --contains--> Central and Cyclic Sources of Normal Subgroups
Normal Subgroups --contains--> Definition and First Examples of Normal Subgroups
Normal Subgroups --contains--> Index Two and Coset Criteria for Normality
Normal Subgroups --contains--> Normalisers of Subgroups
Normal Subgroups --contains--> Normality Inside Related Subgroups
Normal Subgroups --contains--> Products and Intersections of Normal Subgroups
Subgroups and Normal Subgroups --contains--> Quotient Groups
Quotient Groups --contains--> Abelian and Cyclic Quotient Groups
Quotient Groups --contains--> Abelian Quotients and Commutators
Quotient Groups --contains--> Construction of Quotient Groups
Quotient Groups --contains--> Normal Subgroups in Quotient Groups
Quotient Groups --contains--> Order of a Quotient Group
Quotient Groups --contains--> Quotients by Central Subgroups
Abstract Algebra --contains--> Homomorphisms and Isomorphisms of Groups
Homomorphisms and Isomorphisms of Groups --contains--> Homomorphisms of Groups
Homomorphisms of Groups --contains--> Homomorphism of Groups
Homomorphisms of Groups --contains--> Basic Properties of Homomorphisms
Homomorphisms of Groups --contains--> Kernel of a Homomorphism
Homomorphisms of Groups --contains--> Epimorphism, Monomorphism, Isomorphism, and Automorphism
Homomorphisms of Groups --contains--> Natural Homomorphism
Homomorphisms of Groups --contains--> Isomorphism Tests and Examples
Homomorphisms of Groups --contains--> Classification of Cyclic Groups
Homomorphisms of Groups --contains--> Homomorphisms Between Cyclic Groups
Homomorphisms of Groups --contains--> Power Maps on Groups
Homomorphisms of Groups --contains--> Cayley Theorem and Permutation Groups
Homomorphisms and Isomorphisms of Groups --contains--> Isomorphism and Correspondence Theorems
Isomorphism and Correspondence Theorems --contains--> Factorization Through Quotients
Isomorphism and Correspondence Theorems --contains--> First Isomorphism Theorem
Isomorphism and Correspondence Theorems --contains--> Homomorphic Images and Normal Subgroups
Isomorphism and Correspondence Theorems --contains--> Second Isomorphism Theorem
Isomorphism and Correspondence Theorems --contains--> Third Isomorphism Theorem
Isomorphism and Correspondence Theorems --contains--> Correspondence Theorem
Homomorphisms and Isomorphisms of Groups --contains--> Group Actions
Group Actions --contains--> Group Actions and G-Sets
Group Actions --contains--> Orbits and Stabilizers
Group Actions --contains--> Orbit-Stabilizer Formula and Orbit Counting
Group Actions --contains--> Permutation Representations and Extended Cayley Theorem
Group Actions --contains--> Fixed Points and Burnside's Lemma
Abstract Algebra --contains--> Direct Product of Groups
Direct Product of Groups --contains--> External Direct Product
Direct Product of Groups --contains--> Internal Direct Product
Abstract Algebra --contains--> Sylow Theorems
Sylow Theorems --contains--> Conjugacy Classes
Conjugacy Classes --contains--> Centralizers and Conjugacy of Elements
Conjugacy Classes --contains--> Class Equation and Conjugacy Class Decomposition
Conjugacy Classes --contains--> Conjugate Subgroups and Invariance
Conjugacy Classes --contains--> Normalizers and Counting Conjugate Subgroups
Sylow Theorems --contains--> Cauchy's Theorem and p Groups
Cauchy's Theorem and p Groups --contains--> Cauchy's Theorem
Cauchy's Theorem and p Groups --contains--> Subgroups of Finite Abelian Groups
Cauchy's Theorem and p Groups --contains--> p-Groups and p-Subgroups
Cauchy's Theorem and p Groups --contains--> Centers of Finite p-Groups
Cauchy's Theorem and p Groups --contains--> Groups of Order p Squared and p Cubed
Cauchy's Theorem and p Groups --contains--> Subgroups of Finite p-Groups
Cauchy's Theorem and p Groups --contains--> Normal Subgroups from Cauchy's Theorem
Cauchy's Theorem and p Groups --contains--> Computing p-Subgroups
Sylow Theorems --contains--> Sylow Theorems
Sylow Theorems --contains--> Sylow's First Theorem
Sylow Theorems --contains--> Sylow p-Subgroups
Sylow Theorems --contains--> Conjugates and Normality of Sylow p-Subgroups
Sylow Theorems --contains--> Sylow's Second Theorem
Sylow Theorems --contains--> Sylow's Third Theorem
Sylow Theorems --contains--> Sylow Subgroups in Normal Subgroups and Quotients
Sylow Theorems --contains--> Applications of Sylow Counting
Sylow Theorems --contains--> Direct Products of Sylow Subgroups
Sylow Theorems --contains--> Some Applications of the Sylow Theorems
Some Applications of the Sylow Theorems --contains--> Simple Groups and Non-Simplicity Tests
Some Applications of the Sylow Theorems --contains--> Non-Simplicity of Groups of Small Order
Some Applications of the Sylow Theorems --contains--> Simple Groups of Order 60 and A5
Some Applications of the Sylow Theorems --contains--> Groups of Order pq
Some Applications of the Sylow Theorems --contains--> Dihedral Groups and Groups of Order 2p
Some Applications of the Sylow Theorems --contains--> Groups of Small Order
Some Applications of the Sylow Theorems --contains--> Cyclicity from Sylow Subgroups
Some Applications of the Sylow Theorems --contains--> Central Sylow Subgroups and Normality Applications
Some Applications of the Sylow Theorems --contains--> True-or-False Applications of Sylow Theorems
Abstract Algebra --contains--> Solvable and Nilpotent Groups
Solvable and Nilpotent Groups --contains--> Solvable Groups
Solvable and Nilpotent Groups --contains--> Nilpotent Groups
Abstract Algebra --contains--> Finitely Generated Abelian Groups
Finitely Generated Abelian Groups --contains--> Finite Abelian Groups
Finitely Generated Abelian Groups --contains--> Structure Theorem for Finitely Generated Abelian Groups
Abstract Algebra --contains--> Introduction to Rings
Introduction to Rings --contains--> Definition and Basic Properties of Rings
Introduction to Rings --contains--> Important Classes of Rings
Abstract Algebra --contains--> Subrings, Ideals, and Homomorphisms
Subrings, Ideals, and Homomorphisms --contains--> Subrings and Subfields
Subrings, Ideals, and Homomorphisms --contains--> Ideals and Quotient Rings
Subrings, Ideals, and Homomorphisms --contains--> Ring Homomorphisms and Isomorphisms
Abstract Algebra --contains--> Ring Embeddings
Ring Embeddings --contains--> Embedding of Rings
Abstract Algebra --contains--> Direct Sum of Rings
Direct Sum of Rings --contains--> Direct Sum and Product of Rings
Abstract Algebra --contains--> Polynomial Rings
Polynomial Rings --contains--> Polynomial Rings over Fields and Rings
Abstract Algebra --contains--> Euclidean Domains
Euclidean Domains --contains--> Euclidean Domains and Division Algorithm
Euclidean Domains --contains--> Greatest Common Divisors in Domains
Euclidean Domains --contains--> Prime and Irreducible Elements
Abstract Algebra --contains--> Unique Factorization Domains
Unique Factorization Domains --contains--> Unique Factorization Domains
Unique Factorization Domains --contains--> Factorization of Polynomials over UFD
Unique Factorization Domains --contains--> Irreducibility Criteria for Polynomials
Abstract Algebra --contains--> Maximal, Prime, and Primary Ideals
Maximal, Prime, and Primary Ideals --contains--> Maximal, Prime, and Primary Ideals
Maximal, Prime, and Primary Ideals --contains--> Jacobson Semisimple Rings
Abstract Algebra --contains--> Noetherian and Artinian Rings
Noetherian and Artinian Rings --contains--> Noetherian and Artinian Rings
Abstract Algebra --contains--> Modules and Vector Spaces
Modules and Vector Spaces --contains--> Modules and Vector Spaces
Abstract Algebra --contains--> Rings of Matrices
Rings of Matrices --contains--> Full Matrix Rings
Rings of Matrices --contains--> Triangular Matrix Rings
Abstract Algebra --contains--> Field Extensions
Field Extensions --contains--> Field Extensions and Degree
Field Extensions --contains--> Splitting Fields
Field Extensions --contains--> Algebraically Closed Fields
Abstract Algebra --contains--> Multiplicity of Roots
Multiplicity of Roots --contains--> Multiplicity of Roots
Abstract Algebra --contains--> Finite Fields
Finite Fields --contains--> Finite Fields
Abstract Algebra --contains--> Galois Theory and Applications
Galois Theory and Applications --contains--> Normal Extensions
Galois Theory and Applications --contains--> Galois Theory
Galois Theory and Applications --contains--> Roots of Unity and Cyclotomic Polynomials
Galois Theory and Applications --contains--> Solvability of Polynomials by Radicals
Abstract Algebra --contains--> Geometric Constructions
Geometric Constructions --contains--> Geometric Constructions
Abstract Algebra --contains--> Coding Theory
Coding Theory --contains--> Binary Codes
Coding Theory --contains--> Polynomial and Cyclic Codes
Coding Theory --contains--> Bose-Chaudhuri-Hocquenghem Codes
Abstract Algebra --contains--> Gröbner Bases
Gröbner Bases --contains--> Affine Varieties
Gröbner Bases --contains--> Gröbner Bases

Abstract Algebra

Learning structure

Abstract Algebra Learning Map

Concepts

Relationships

Chapters

Sets, Relations, and Integers

Introduction to Groups

Permutation Groups

Subgroups and Normal Subgroups

Homomorphisms and Isomorphisms of Groups

Direct Product of Groups

Sylow Theorems

Solvable and Nilpotent Groups

Finitely Generated Abelian Groups

Introduction to Rings

Subrings, Ideals, and Homomorphisms

Ring Embeddings

Direct Sum of Rings

Polynomial Rings

Euclidean Domains

Unique Factorization Domains

Maximal, Prime, and Primary Ideals

Noetherian and Artinian Rings

Modules and Vector Spaces

Rings of Matrices

Field Extensions

Multiplicity of Roots

Finite Fields

Galois Theory and Applications

Geometric Constructions

Coding Theory

Gröbner Bases

Curated pathways

Metric Spaces

Introduction To Discrete Mathematics