Abstract Algebra

  1. Field Theory and Galois Theory
    1. Introduction to Fields
      1. Review of Field Axioms
        1. Additive Structure
          1. Multiplicative Structure
          2. Characteristic of a Field
            1. Definition and Examples
            2. Prime Fields
              1. Construction and Properties
                1. Q and Z_p as Prime Fields
              2. Field Extensions
                1. Base Field and Extension Field
                  1. Definitions and Examples
                  2. Degree of an Extension
                    1. Definition and Calculation
                      1. Finite and Infinite Extensions
                      2. The Tower Law
                        1. Statement and Applications
                          1. Multiplicativity of Degrees
                        2. Algebraic and Transcendental Extensions
                          1. Algebraic Elements
                            1. Definition and Examples
                            2. Transcendental Elements
                              1. Definition and Examples
                              2. Minimal Polynomial of an Algebraic Element
                                1. Existence and Uniqueness
                                2. Simple Extensions
                                  1. Definition and Properties
                                    1. F(α) Construction
                                    2. Finitely Generated Extensions
                                      1. Definition and Examples
                                    3. Straightedge and Compass Constructions
                                      1. Constructible Numbers
                                        1. Definition and Properties
                                          1. Field of Constructible Numbers
                                          2. Impossibility of Classical Problems
                                            1. Squaring the Circle
                                              1. Doubling the Cube
                                                1. Trisecting the Angle
                                              2. Splitting Fields and Algebraic Closure
                                                1. Definition of a Splitting Field
                                                  1. Construction and Examples
                                                  2. Existence and Uniqueness of Splitting Fields
                                                    1. Proof Techniques
                                                    2. Algebraically Closed Fields
                                                      1. Definition and Examples
                                                      2. The Fundamental Theorem of Algebra
                                                        1. Statement and Consequences
                                                      3. Finite Fields
                                                        1. Existence and Uniqueness
                                                          1. Construction of Finite Fields
                                                            1. Galois Fields GF(p^n)
                                                            2. Structure of Finite Fields
                                                              1. Multiplicative Group
                                                                1. Additive Group
                                                                2. Subfields of a Finite Field
                                                                  1. Classification and Examples
                                                                    1. Lattice of Subfields
                                                                  2. Galois Theory
                                                                    1. The Galois Group of an Extension
                                                                      1. Definition and Examples
                                                                        1. Automorphisms of Field Extensions
                                                                        2. Separable Extensions
                                                                          1. Definition and Properties
                                                                            1. Separable Polynomials
                                                                            2. Normal Extensions
                                                                              1. Definition and Properties
                                                                                1. Splitting Fields as Normal Extensions
                                                                                2. Galois Extensions
                                                                                  1. Definition and Properties
                                                                                    1. Finite Separable Normal Extensions
                                                                                    2. The Fundamental Theorem of Galois Theory
                                                                                      1. Statement and Applications
                                                                                        1. Galois Correspondence
                                                                                        2. Correspondence between Subgroups and Subfields
                                                                                          1. Fixed Fields
                                                                                            1. Galois Groups of Subextensions
                                                                                          2. Applications of Galois Theory
                                                                                            1. The Insolvability of the Quintic by Radicals
                                                                                              1. Historical Context
                                                                                                1. Proof Outline
                                                                                                2. Cyclotomic Extensions
                                                                                                  1. Definition and Properties
                                                                                                    1. Cyclotomic Fields
                                                                                                    2. Cyclotomic Polynomials
                                                                                                      1. Definition and Properties
                                                                                                        1. Irreducibility

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                                                                                                    3. Ring Theory

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                                                                                                    5. Module Theory