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Math E-222 - Abstract Algebra

Math E-222 - Abstract Algebra (Fall 2003, Harvard Extension School). Instructor: Professor Benedict Gross. Algebra is the language of modern mathematics. This course introduces students to that language through a study of groups, group actions, vector spaces, linear algebra, and the theory of fields. Topics include: Review of Linear Algebra; Permutations; Quotient Groups, First Isomorphism Theorem; Abstract Linear Operators and How to Calculate with Them; Orthogonal Groups; Isometrics of Plane Figures; Group Actions; A5 and the Symmetries of an Icosahedron; Rings; Euclidean Domains, PIDs, UFDs; and Structure of Ring of Integers in a Quadratic Field.

Introduction


Review of Linear Algebra
Lecture 01 - Introduction to the Course; Review: Linear Algebra; Definition of Groups
Lecture 02 - Generalities on Groups; Examples of Groups
Lecture 03 - Isomorphisms; Homomorphisms; Images
Permutations
Lecture 04 - Review, Kernels, Normality; Examples; Centers and Inner Autos
Lecture 05 - Equivalence Relations; Cosets; Examples
Lecture 06 - Congruence Mod n; (Z/nZ)*
Quotient Groups, First Isomorphism Theorem
Lecture 07 - Quotients
Lecture 08 - More on Quotients; Vector spaces
Lecture 09 - Vector spaces (continued)
Abstract Linear Operators and How to Calculate with Them
Lecture 10 - Bases and Vector spaces; Matrices and Linear Transformations
Lecture 11 - Bases; Matrices
Lecture 12 - Eigenvalues and Eigenvectors
Lecture 13 - Review for Midterm; Orthogonal Group
Orthogonal Groups
Lecture 14 - Orthogonal Group and Geometry
Lecture 15 - Finite Groups of Motions
Lecture 16 - Discrete Groups of Motions
Isometrics of Plane Figures
Lecture 17 - Discrete Groups of Motions; Abstract Group Actions
Lecture 18 - Group Actions
Lecture 19 - Group Actions (continued)
Group Actions
Lecture 20 - Group Actions: Sylow Theorems
Lecture 21 - Group Actions: Sylow Theorems (continued), Classification Theorems
Lecture 22 - Group Actions: The Symmetric Group, Conjugation, S5 Classes
A5 and the Symmetries of an Icosahedron
Lecture 23 - Alternating Group Structure
Lecture 24 - Rings
Lecture 25 - Rings (continued)
Rings
Lecture 26 - R Commutative Ring, Quotient Rings and Isomorphisms
Lecture 27 - Examples of Rings
Lecture 28 - Rings: Review
Extensions of Rings
Lecture 29 - Quotient Rings, Integral Domains, Fields of Fractions
Special Lecture
Lecture 30 - Domains & Factorization in Z, Euclidean Algorithm
Lecture 31 - Domains & Factorization in Z (cont.), Gauss' Lemma
Lecture 32 - Gaussian Integers
Euclidean Domains, PIDs, UFDs
Lecture 33 - Gauss' Lemma, Eisenstein's Criterion, Algebraic Integers
Lecture 34 - Gauss' Lemma, Eisenstein's Criterion, Algebraic Integers (cont.)
Lecture 35 - Prime and Maximal Ideals, Dedekind Domains, Class Groups
Structure of Ring of Integers in a Quadratic Field
Lecture 36 - Dedekind Domains, Ideal Class Groups
Review
Lecture 37 - Review 1
Lecture 38 - Review 2

References
Abstract Algebra
Instructor: Professor Benedict Gross. Lecture Notes. Problem Sets. This course introduces students to that language through a study of groups, group actions, vector spaces, linear algebra, and the theory of fields.