# InfoCoBuild

## 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.