Course Descriptions

16:640:551 - Abstract Algebra I

Fall 2023

Simon Thomas

Course Description:

This is a standard course for beginning graduate students. It covers Group Theory, basic Ring & Module theory, and bilinear forms. Group Theory: Basic concepts, isomorphism theorems, normal subgroups, Sylow theorems, direct products and free products of groups. Groups acting on sets: orbits, cosets, stabilizers. Alternating/Symmetric groups. Basic Ring Theory: Fields, Principal Ideal Domains (PIDs), matrix rings, division algebras, field of fractions. Modules over a PID: Fundamental Theorem for abelian groups, application to linear algebra: rational and Jordan canonical form. Bilinear Forms: Alternating and symmetric forms, determinants. Modules: Artinian and Noetherian modules. Krull-Schmidt Theorem for modules of finite length. Simple modules and Schur's Lemma, semisimple modules. Finite-dimensional algebras: Simple and semisimple algebras, Artin-Wedderburn Theorem, group rings, Maschke's Theorem.

Text:

Abstract Algebra, Dummit and Foote

Prerequisites:

Standard course in Abstract Algebra for undergraduate students

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Fall 2022

Anders Buch

Course Description:

This will be a basic graduate algebra course covering categories, group theory, rings, fields, and modules, including multilinear algebra, Galois theory, and modules over principal ideal domains. Category theory will help to organize the material into a common framework.

Text:

Brundan and Kleshchev

Prerequisites:

Familiarity with basic algebra such as polynomials, vector spaces, etc.

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Fall 2021 

Jerrold Tunnell

Course Description:

This will be a basic graduate algebra course discussing such algebraic structures as categories, monoids and groupoids, groups, rings, and modules. Since categories form a basic structure having many applications as diverse as physics or computer science, we will begin there. Examples of algebraic objects will be emphasized throughout, as well as connections with other areas of mathematics. The first three chapters of the text will form the kernel of the course, with additional topics introduced in the lectures.

1. Categories and functors

2. Groups and homomorphisms

3. Rings, ideals and principal ideal domains

4. Modules, homomorphisms and structure theorems

Course Format: There will be weekly homework assignments, as well as longer term projects.

Text:

Jacobson, BasicAlgebra I

Prerequisites:

Familiarity with basic algebra such as polynomials, vector spaces, etc.

Schedule of Sections:

 

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