Math E222  Abstract Algebra
Math E222  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.
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, S_{5} 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.
