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Math E-222 - Abstract Algebra (Fall 2003, Harvard Extension School). Instructor: Professor Benedict Gross. Algebra is the language of modern mathematics.
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English [CC]
FREE
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Description
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.
Course content
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- Lecture 01 – Introduction to the Course; Review: Linear Algebra; Definition of Groups Unlimited
- Lecture 02 – Generalities on Groups; Examples of Groups Unlimited
- Lecture 03 – Isomorphisms; Homomorphisms; Images Unlimited
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- Lecture 04 – Review, Kernels, Normality; Examples; Centers and Inner Autos Unlimited
- Lecture 05 – Equivalence Relations; Cosets; Examples Unlimited
- Lecture 06 – Congruence Mod n; (Z/nZ)* Unlimited
- Lecture 07 – Quotients Unlimited
- Lecture 08 – More on Quotients; Vector spaces Unlimited
- Lecture 09 – Vector spaces (cont.) Unlimited
- Lecture 14 – Orthogonal Group and Geometry Unlimited
- Lecture 15 – Finite Groups of Motions Unlimited
- Lecture 16 – Discrete Groups of Motions Unlimited
- Lecture 20 – Group Actions: Sylow Theorems Unlimited
- Lecture 21 – Group Actions: Sylow Theorems (continued), Classification Theorems Unlimited
- Lecture 22 – Group Actions: The Symmetric Group, Conjugation, S5 Classes Unlimited
- Lecture 26 – R Commutative Ring, Quotient Rings and Isomorphisms Unlimited
- Lecture 27 – Examples of Rings Unlimited
- Lecture 28 – Rings: Review Unlimited
- Lecture 30 – Domains and Factorization in Z, Euclidean Algorithm Unlimited
- Lecture 31 – Domains and Factorization in Z (cont.), Gauss’ Lemma Unlimited
- Lecture 32 – Gaussian Integers Unlimited
- Lecture 36 – Dedekind Domains, Ideal Class Groups Unlimited
Instructor
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