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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.

Lecture 26 - R Commutative Ring, Quotient Rings and Isomorphisms

Go to the Course Home or watch other lectures:

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 (cont.)

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 (cont.)

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 (cont.)

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 and Factorization in Z, Euclidean Algorithm

Lecture 31 - Domains and 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