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.
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 |