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/Zoom
Office hours: (Zoom) Mondays 11am-12pm, Wed 4-5pm or by appointment
Teaching assistant: Iris Rosenblum-Sellers, irg2102 (at) Columbia (dot) edu  
TA office hours: (Zoom) Mondays 4-5pm, Wed 5-6pm

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.       Homework 2, due Wed Sept 23.      

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

MathDoctorBob: Definition of a ring   Ring homomorphisms   Definition of integral domain   Example of integral domain

Weeks 3-4:
Lecture 4 slides, Monday Sept 21.     Lecture 5 slides, Wed Sept 23.     Lecture 6 slides, Monday Sept 28.     Lecture 7 slides, Monday Sept 30.    

Homework 3, due Wed Sept 30.       Homework 4, due Wed October 7.      

Notes by Robert Friedman: Ideals   Factorizations in polynomial rings   Euclidean algorithm (for integers)

MathDoctorBob: Ideals and quotient rings  

Weeks 5-6:
Lecture 8 slides, Wed Oct 7.     Lecture 9 slides   Notes, Mon Oct 12.     Lecture 10 slides,   Notes, Wed Oct 14.    

Homework 5, due Wed Oct 14.       Homework 6, due Wed Oct 21.

Midterm 1, Oct 5     Midterm 1, earlier version

Notes by Robert Friedman: Linear algebra over fields   Field extensions 1   Multiple roots   Finite fields

MathDoctorBob: Characteristic p   Field extensions  

Weeks 7-8:
Lecture 11 slides,   Notes, Mon Oct 19.     Lecture 12 slides,   Notes, Wed Oct 21.

Lecture 13 slides,   Notes, Mon Oct 26.     Lecture 14 slides,   Notes, Wed Oct 28.

Homework 7, due Wed Oct 28.     Homework 8, due Wed Nov 4.    

Notes by Robert Friedman: Galois Theory I   II   III  IV

Weeks 9-10:
Lecture 15 slides,   Notes, Wed Nov 4.    

Lecture 16 slides,   Notes, Mon Nov 9.     Lecture 17 slides,   Notes, Wed Nov 11.

Homework 9, due Wed Nov 11.     Homework 10, due Wed Nov 18.

Weeks 11-12:
Lecture 18 Notes, Monday Nov 16.     Ruler-Compass constructions: Rotman     Morandi (Field and Galois theory, full book available via Columbia online library).

Lecture 19 slides,   Notes, Wed Nov 18.     Optional reading: Morandi, cyclotomic extensions.

Homework 11, due Wed Nov 25.

Weeks 13-14:
Lecture 20 slides,   Notes, Mon Nov 30.     Lecture 21 slides,   Notes, Wed Dec 2.

Homework 12, due Wed Dec 9.

Lecture 22 slides,   Notes, Mon Dec 7.     Lecture 23 slides,   Notes, Wed Dec 9.    

Notes by Robert Friedman: Chinese Remainder Theorem

Week 15:
Lecture 24 notes

Review Session Part I    Part II

Additional resources:
Robert Donley (MathDoctorBob on Youtube) has an online course on Modern Algebra.

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.