Columbia University GU4042
Introduction to Modern Algebra II, fall 2020

Basic information

Call number: 11488
Time/Place: MW 1:10pm--2:25pm, online
Instructor: Mikhail Khovanov
Office: 620 Math or online
Office hours: (first week) Friday 11am-12pm or by appointment
Teaching assistant:    TBA  

Textbook: Galois Theory, by Joseph Rotman, second edition (1998). You can get pdf file from Columbia Online Library (follow "SpringerLink ebooks" link on the right) as well as purchase a printed copy from Springer via MyCopy service on the same webpage as the pdf download.

Additional textbooks: There are many excellent textbooks that cover similar material.

This is the second semester of a 2-semester course on Modern Algebra. The first semester, which covered group theory, is the prerequisite for this course.

Syllabus: Rings and commutative rings. Rings of polynomials, residues modulo n and other examples. Matrix rings and quaternions. Integral domains and fields. Field of fractions. Homomorphisms of rings and ideals. Quotient rings and First Isomorphism Theorem for rings. Principal ideal domains and polynomial rings over fields. Prime and maximal ideals. Irreducible polynomials. Characteristic of a field. Finite fields. Linear algebra over a field. Field extensions and splitting fields. Galois group. Solvability by Radicals. Ruler and compass constructions. Independence of characters. Galois' Theorems. Applications. Fundamental Theorem of Algebra. Applications of finite fields.
If time allows: Modules over rings and representation theory. Classification of (finitely-generated) modules over PIDs. Semisimple rings. Basics of category theory.

Homework: Homework will be assigned on Wednesdays, due Wednesday the next week before class. It will be posted on this webpage. The first problem set is due September 16. The lowest homework score will be dropped. You can discuss homework problems with your fellow students, after you make a serious effort to solve each problem on your own. Homework discussion prior to submission is subject to the following rules: (1) List the name of your collaborators at the head of the problem or assignment, (2) Do not exchange written work with others, (3) Write up solutions in your own words.
Throughout the semester we'll have several 10-minute quizzes, with yes/no and multiple choice questions.

The numerical grade for the course will be the following linear combination: 5% quizzes, 20% homework, 20% each midterm, 35% final.

Weeks 1-2:
Lecture 1 slides, Wed Sept 9.     Lecture 2 slides, Mon Sept 14.     Lecture 3 slides, Wed Sept 16.

Homework 1, due Wed Sept 16.      

Supplemental resources for weeks 1-2:
Notes by Robert Friedman: Rings   Polynomials   Integral domains

Robert Donley (MathDoctorBob on Youtube) has an online course on Modern Algebra.
Videos Definition of a ring   Ring homomorphisms   Definition of integral domain   Example of integral domain relate to the material we cover in lectures 1-3.

Weeks 3-4:
MathDoctorBob: Ideals and quotient rings  

Other algebra texts: There are many that you can find online or in the library. A rather incomplete list: Michael Artin Algebra, John Fraleigh A First Course in Abstract Algebra, Joseph Gallian Contemporary Abstract Algebra, Thomas Hungerford Abstract Algebra: An Introduction, Serge Lang Undergraduate Algebra. Dummit and Foote Abstract Algebra is truly encyclopedic without losing textbook qualities, a popular graduate school textbook.