601 Algebra 2

• Modules
• Constructions On Modules
• Free Modules
• Exact Sequences
• Tensor Product Exposition
• Constructing Tensor Product
• Hom Tensor Duality
• Properties Of Tensor Product
• Examples Of Tensor Products
• Tensor Product Of Free Modules
• Tensor Algebra
• Symmetric Algebra
• Applications Of Symmetric Power
• Exterior Algebra
• Functorality Of Alternating Power
• Determinant
• Properties Of Determinants
• Localization
• Torsion Sub module
• Torsion Free Over PID Is Free
• Structure Of Sub modules Of Free Modules Over PID
• Structure Of Sub modules Continued
• Invariant Divisor And Elementary Factor Form
• Minimal Polynomial
• Rational Canonical Form
• More Rational Canonical Form
• Jordan Canonical Form
• Constructing Jordan Canonical Form
• Categories
• Functors
• Natural Transformation
• Exact Functors
• Projective Modules
• Injective Z Modules
• Injective A Modules
• Injective Modules And Short Exact Sequence
• Tensor Functor Is Right Exact
• Flat Modules
• Equivalence Of Categories
• Vector Bundles
• K Theory
• Cohomology
• Long Exact Sequence Of Homology
• Homotopy
• Ext Group
• Mapping Between Resolutions
• Ext Is Well Defined
• Long Exact Sequence In Ext
• More Ext Facts
• Tor Groups
• Field Extensions
• Towers Of Extensions
• Transcendental And Algebraic Extensions
• Minimal Polynomial Of Extension
• Low Degree Extensions
• Algebraic Fields
• Algebraic Subfield
• Towers Of Algebraic Extensions
• Constructions With Straight Edge And Compass
• Impossible Constructions
• Constructing N Gons
• Existence Of Splitting Fields
• More Splitting Field
• Uniqueness Of Splitting Field
• Normal Extensions
• Finite Normal Extensions
• Algebraic Closures
• Embedding Into Algebraically Closed Field
• Algebraic Closure
• Separable Polynomial
• Derivatives Of Polynomial
• Perfect And Separable Fields
• Cyclotomic Polynomials
• Irreducibility Of Cyclotomic Polynomial
• Galois Group
• Galois Extensions
• Major Theorems Of Galois Theory
• Constructing Finite Fields
• Subfields Of Finite Fields
• Finite Multiplicative Subgroups Of Fields
• Galois Group Of Finite Fields
• Theorem Of Primitive Element
• Quadratic Galois Examples
• Examples Of Galois
• More Galois Examples
• Inverse Galois Problem
• Constructing Minimal Polynomials
• Lemmas To Characterize Galois Extensions
• Characterizing Galois Extensions
• Proof Of Fundamental Theorem Of Galois
• Introduction To Solvable By Radicals
• Simple Radical Extension
• Constructing Cyclic Root Extensions
• Composites And Galois Groups
• Galois And Abel Theorems
• Linear Representation
• Subrepresentations
• Irreducible Subrepresentations
• Isomorphic Representations
• Shur's Lemma
• Abelian Representations
• Constructions With Representations
• Characters Of A Representation
• Inner Product On Class Functions
• Irreducible Representations In Standard Representation
• Character Table Example
• Irreducible Characters Form Basis
• Degree Of Character Of Divides Order Of The Group
• Character Tables
• Character Tables Don't Determine Groups Uniquely
• Kernel Of A Character
• Solvability And Character
• C- Algebras As Representations
• Compact Group Representation
• Theorems Of Compact Group Representation
• Numbering Of Partitions
• Example Representation Of Sn
• Hook Length Formula
• Character Of Representations Of Sn
• Murngahm Nakaymah Rule
• Polynomial Representations Of GL(n,C)
• Schur Modules
• Character Of Schur Representation
• Untitled