Abstract Algebra

  1. Ring Theory
    1. Introduction to Rings
      1. Definition and Axioms of a Ring
        1. Addition Operation
          1. Multiplication Operation
            1. Ring Axioms
            2. Basic Properties
              1. Zero Element
                1. Additive Inverses
                  1. Distributive Laws
                  2. Commutative Rings
                    1. Definition and Examples
                    2. Noncommutative Rings
                      1. Definition and Examples
                      2. Rings with Unity
                        1. Definition and Examples
                          1. Units in a Ring
                        2. Examples of Rings
                          1. Rings of Numbers
                            1. Integers
                              1. Rational Numbers
                                1. Real Numbers
                                  1. Complex Numbers
                                  2. The Ring of Integers Modulo n
                                    1. Construction and Properties
                                    2. Rings of Polynomials
                                      1. Polynomial Rings over Fields
                                        1. Polynomial Rings over Rings
                                        2. Rings of Matrices
                                          1. Matrix Addition and Multiplication
                                          2. Rings of Functions
                                            1. Pointwise Operations
                                              1. Continuous Functions
                                            2. Special Types of Rings
                                              1. Integral Domains
                                                1. Definition and Properties
                                                  1. Zero Divisors
                                                  2. Division Rings
                                                    1. Definition and Examples
                                                    2. Fields
                                                      1. Definition and Properties
                                                        1. Relationship to Other Ring Types
                                                      2. Subrings and Ideals
                                                        1. Definition of a Subring
                                                          1. Subring Test
                                                          2. Definition of an Ideal
                                                            1. Left Ideals
                                                              1. Right Ideals
                                                                1. Two-Sided Ideals
                                                                2. Principal Ideals
                                                                  1. Definition and Examples
                                                                    1. Principal Ideal Domains
                                                                    2. Operations on Ideals
                                                                      1. Sum of Ideals
                                                                        1. Product of Ideals
                                                                          1. Intersection of Ideals
                                                                        2. Quotient Rings and Homomorphisms
                                                                          1. Ring Homomorphisms
                                                                            1. Definition and Examples
                                                                              1. Properties of Homomorphisms
                                                                              2. Kernel of a Homomorphism
                                                                                1. Definition and Properties
                                                                                2. Image of a Homomorphism
                                                                                  1. Definition and Properties
                                                                                  2. Quotient Rings
                                                                                    1. Construction and Properties
                                                                                      1. Well-Definedness
                                                                                      2. The Isomorphism Theorems for Rings
                                                                                        1. First Isomorphism Theorem
                                                                                          1. Second Isomorphism Theorem
                                                                                            1. Third Isomorphism Theorem
                                                                                          2. Prime and Maximal Ideals
                                                                                            1. Definition of a Prime Ideal
                                                                                              1. Properties and Examples
                                                                                                1. Characterizations
                                                                                                2. Definition of a Maximal Ideal
                                                                                                  1. Properties and Examples
                                                                                                    1. Characterizations
                                                                                                    2. Relationship between Ideals and Quotient Rings
                                                                                                      1. Prime Ideals and Integral Domains
                                                                                                        1. Maximal Ideals and Fields
                                                                                                      2. Polynomial Rings
                                                                                                        1. Polynomials over a Ring R[x]
                                                                                                          1. Definitions and Notation
                                                                                                            1. Degree of a Polynomial
                                                                                                            2. The Division Algorithm for Polynomials
                                                                                                              1. Statement and Applications
                                                                                                                1. Conditions for Validity
                                                                                                                2. Roots of Polynomials
                                                                                                                  1. Definition and Properties
                                                                                                                    1. Factor Theorem
                                                                                                                    2. Reducibility and Irreducibility
                                                                                                                      1. Definitions and Criteria
                                                                                                                        1. Irreducible Polynomials over Fields
                                                                                                                        2. Eisenstein's Criterion
                                                                                                                          1. Statement and Applications
                                                                                                                        3. Factorization in Commutative Rings
                                                                                                                          1. Units in a Ring
                                                                                                                            1. Definition and Properties
                                                                                                                            2. Associates in a Ring
                                                                                                                              1. Definition and Properties
                                                                                                                              2. Irreducible Elements
                                                                                                                                1. Definition and Properties
                                                                                                                                2. Prime Elements
                                                                                                                                  1. Definition and Properties
                                                                                                                                  2. Unique Factorization Domains
                                                                                                                                    1. Definition and Examples
                                                                                                                                    2. Principal Ideal Domains
                                                                                                                                      1. Definition and Properties
                                                                                                                                        1. Relationship to UFDs
                                                                                                                                        2. Euclidean Domains
                                                                                                                                          1. Definition and Examples
                                                                                                                                            1. Euclidean Algorithm
                                                                                                                                            2. Relationships between Euclidean Domains, PIDs, and UFDs
                                                                                                                                              1. Hierarchy of Ring Types

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                                                                                                                                          4. Field Theory and Galois Theory