600 Algebra 1

• Basic Notation
• Vector Spaces
• Subspaces And Span
• Linear Independence
• Basis
• Dimension
• More Dimension
• Zorns Lemma And Axiom Of Choice
• Basis Of Any Vector Space
• Linear Trabsformations
• Kernel And Image Of Linear Maps
• Isomorphism Of Vector Space
• Direct Sum Of Vector Space
• Compliments And Direct Products Of Vector Spaces
• Quotient Vector Space
• Quotient Vector Space
• Isomorphism Theorems
• Universal Properties
• Hom Spaces
• Coordinate Systems
• Classifying Linear Maps Up To Two Change Of Basis
• Eigenvalues
• Diagonalizable Maps
• General Form Of C Linear Maps
• Inner Product Space
• Orthogonal
• Orthogonal Transforms
• Rigid Motions
• Groups
• Subgroups
• Generating Subgroups
• Cosets Of Subgroups
• Modular Arithmetic
• Quotient Groups
• Group Homomorphsims
• Direct Product
• Automorphisms Of Groups
• Symmetries Of Rn
• Plane Symmetries
• Higher Dimensional Symmetry
• Group Action
• Orbits Of Group Action
• Stabilizer Of Group Action
• Centralizer And Class Equation
• Actions On Subgroups
• Symmetric Group
• Alternating Group
• Conjugacy Classes Of Sn
• Families Of Simple Groups
• Classifications Of Finite Simple Groups
• First Sylow Theorem
• Second And Third Sylow Therms
• Composition Series
• Internal Semidirect Product
• External Semidirect Product
• More Semi Direct Product
• Free Groups
• Group Presentations
• Cayley Graph
• Adjoint Of Linear Map
• Self Adjoint Linear Maps
• Spectral Theorem For Self Adjoint Maps
• Quadratic Forms
• Sign Of Quadratic Forms
• Generalized Second Derivative Test
• Rings
• More Rings
• Integral Domains
• Ring Homomorphism
• Ideals And Quotients
• Generating Ideals
• Maximal Ideals
• Prime Ideals
• Field Of Fractions
• Adjoining Elements To Rings
• Euclidean Domains
• Polynomial Division
• Principle Ideal Domain
• Euclidean Algorithm
• Prime And Irreducible Elements
• Unique Factorization Domains
• Equivalent Characterizations Of A UFD
• Relations Of UFD To Other Domains
• Gauss Lemma
• Corollary To Gauss Lemma
• Gauss Theorem
• Criteria For Irreducible In Polynomial
• Eisensteins Criteria
• Chinese Remainder Theorem
• Noetherian Rings
• Hilbert Basis Theorem
• Introduction To Algebraic Geometry
• Algebraic Sets
• Radical Ideals
• Irreducible Algebraic Sets
• Classifying Low Dimensional Irreducible Algebraic Sets
• Weak Hilbert Nullstellensatz
• Strong Nullstellensatz
• Affine Variety
• Constructing Tensor Product
• Hom Tensor Duality