List of abstract algebra topics

Basic language

• Binary operation
  • Multiplication table
• Commutative operation
• Identity element
• Additive inverse, multiplicative inverse, inverse element
• Associativity
• Distributivity
  • Distributive property
• Semigroup, monoid

Group theory

• Group (mathematics)
• Finite group
• Coset
• Lagrange's theorem
• Abelian group
  • Torsion subgroup
  • Direct sum of groups
  • Free abelian group
  • Finitely generated abelian group
  • Rank of an abelian group
  • Divisible group
  • Locally cyclic group
• Cyclic group
• Klein four-group
• Elementary group theory
  • Quaternion group
  • List of small groups
  • Examples of groups
• Normal subgroup
  • Conjugacy class
    • Inner automorphism
    • Conjugate closure
  • Semidirect product
  • Dihedral group
  • Dicyclic group
  • Extension problem
  • Hamiltonian group
  • Simple group
    • Classification of finite simple groups
  • Composition series
  • Solvable group
  • Nilpotent group
• Characteristic subgroup
• Derived group
• Centralizer and normalizer
• Automorphism group
• Sylow theorem
  • Local analysis
• Group action
  • Permutation group
  • Symmetric group
  • Alternating group
  • Stabilizer subgroup
  • Orbit (mathematics)
  • Orbit-stabilizer theorem
  • Cayley's theorem
  • Burnside's lemma
  • Wreath product
• Free group
• Presentation of a group
• Free product
• Word problem for groups
• Burnside's problem

Ring theory

• Ring (mathematics)
• Subring
• Ring homomorphism
• Ring ideal
  • Principal ideal
  • Maximal ideal, prime ideal
  • Jacobson radical
  • Radical of an ideal
• Product of rings
• Idempotent
• Nilpotent
• Zero divisor
• Integral domain
• Euclidean domain
• Principal ideal domain
• Field (mathematics)
• Field of quotients
• Commutative algebra
• Polynomial, monomial
• Center (algebra)
• Integral closure
• Dedekind domain
  • Ideal class group
• Unique factorization domain
• Formal power series
• Characteristic
• Boolean algebra, Boolean ring
• Division ring, division algebra
• Local ring
• Valuation (mathematics)
• Monoid ring
• Group ring
• Group algebra
• Dual number
• Derivation
• Artinian
• Noetherian
  • Hilbert's basis theorem
• Banach algebra
• Associative algebra
  • Subalgebra
  • Algebra over a field
  • Central simple algebra
  • Artin-Wedderburn theorem
  • Tensor algebra
  • Tensor product of R-algebras
  • Brauer group
• Hypercomplex number
  • Graded algebra
• Rig (algebra)

Field theory

• Subfield (mathematics)
• Field extension
• Algebraic number field
• Global field
• Local field
• Finite field
• Algebraic element
• Algebraic extension
• Symmetric function
• Splitting field
• Separable polynomial
• Separable extension
• Primitive element (field theory)
• Algebraic closure
• Algebraically closed field
• Transcendence degree
• Normal extension
• Galois extension
• Galois theory
  • Galois group
  • Conjugate element (field theory)
  • Abelian extension
• Field norm
• Field trace
• Tensor product of fields

Module theory

• Free module
• Finitely-generated module

Referenced By