Columbia University GU4042
Introduction to Modern Algebra II, spring 2022

Basic information

Call number: 11896
Time/Place: MW 2:40pm--3:55pm, Math 312 (via Zoom in January)
Instructor: Mikhail Khovanov
Office: 620 Math (on Zoom in January)
Office hours: (tentative) Wednesday 11am-12pm, Friday 4-5pm or by appointment
Course webpage:
Midterm 1: Wednesday, February 16
Final Exam: Wednesday, May 11, 1:10pm-4pm, room Math 312
Teaching assistant: Nguyen Chi Dung    nguyendung (AT) math (DOT) columbia (DOT) edu,   Office Hours: Monday 10am-1pm in Help Room (Math 406); online January 24.
Teaching assistant: Jacob Weinstein    jmw2281 (AT) columbia (DOT) edu,   Office Hours: Tuesday 4pm-6pm in Help Room (Math 406), online January 25.

Textbooks: We will use two textbooks: (1) Galois Theory, by Joseph Rotman, second edition (1998) and Fields and Galois Theory, by John Howie.
You can get pdf files of both books from Columbia Online Library: Rotman and Howie (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.

The first semester of Modern Algebra (group theory) is the prerequisite for this course.

Syllabus: Rings and commutative rings. Rings of polynomials, residues modulo n and other examples. Matrix rings. 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. Euclidean domains and unique factorization domains. Characteristic of a field. Finite fields. Linear algebra over a field. Field extensions and splitting fields. Field extensions of field automorphisms. 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 January 26. 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: 4% quizzes, 20% homework, 20% each midterm, 36% final.

WEEKS 1-2:
Lecture 1 slides       Lecture 1 notes, Wed Jan 19.

Lecture 2 slides       Lecture 2 notes, Mon Jan 24. Associativity-commutativity origins      

Lecture 3 slides       Lecture 3 notes, Wed Jan 26.

Homework 1, due Wed Jan 26.

Homework 2, due Wed Feb 2.

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

Robert Donley (MathDoctorBob on Youtube) has an online course on Modern Algebra.
Videos Definition of a ring   Ring homomorphisms  

WEEKS 3-4:
Lecture 4   Mon Jan 31,       Lecture 5   Wed Feb 2.      

Homework 3, due Wed Feb 9.       Homework 4, due Wed Feb 14.

Lectures 6, 7 (Feb 7, 9) cover the same material as lectures 5-7 from my 2020 course: Lect 5     Lect 6     Lect 7    

Notes by Robert Friedman: Integral domains   Ideals   Factorization in polynomial rings   Euclidean algorithm (for integers)  

MathDoctorBob: Definition of integral domain   Example of integral domain   Ideals and quotient rings  

WEEKS 5-6:
Homework 5, due Wed Mar 2.    

Lecture 8   Mon Feb 14      
Field from an irreducible polynomial   Ideals in direct product  

Lectures 9, 10 are covered by notes for Lecture 8 and Lecture 9 from 2020 course.

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

MathDoctorBob: Maximal ideals and fields   Finite fields of orders 4 and 8   Characteristic p   Field extensions  

WEEKS 7-9:
Homework 6, due Wed Mar 9.    

Homework 7, due Wed Mar 23.    

Homework 8, due Wed Mar 30.    

For lectures 11, 12 use notes for Lecture 9 and Lecture 10 from 2020 course. In these lectures we continued to use Friedman's notes Field extensions 1 and Finite fields. Alternatively, you can read Howie, Section 3 (Field extensions) and Section 6 (Finite fields).

For lectures 13, 14 see notes for Lecture 11 and Lecture 12 from 2020 course and the above notes by R.Friedman.

Lecture 15 notes, also see Lecture 12 from 2020 course and parts of Galois Theory I notes. In Rotman, see Splitting Fields (pages 50-58) and beginning of Galois Group section (pages 59-61). In Howie, read Section 5 Splitting Fields, Section 7.2 (Automorphisms). You may also take a look at Section 7.3 (Normal extensions) which discusses splitting field extensions using an equivalent notion of a normal extension.

WEEKS 10-11:
Homework 9, due Friday April 8.    

Homework 10, due Wed April 13.    

Lecture 16: Ruler and compass constructions. The most detailed reference is Abstract Algebra and Famous Impossibilities, by A.Jones, S.Morris and K.Pearson, Chapters 5, 6. Also see Howie, Chapter 4 for a much shorter exposition, or Rotman, Appendix C.

Lecture 17: Commutator subgroup, Vandermonde determinant, linear independence of characters.
Vandermonde determinant: MathDoctorBob video, Notes by Paul Garrett.
Commutator subgroup:
MathDoctorBob video, Notes by Ryan Vinroot.

Additional videos of MathDoctorBob that are very close to the material of current and recent lectures:
Automorphisms and degree, Galois Correspondence 1 examples. Videos 65-80 in his playlist Abstract Algebra deal with topics and examples related to Galois groups, field extensions,

Finite fields resources: Chapter 6 of Howie and Friedman's notes on finite fields we posted in Weeks 5-6. Rotman spreads out discussion of finite fields through the book, see pages 36-37 and 65-68. A 4-page summary of finite fields.

WEEKS 12+:
Homework 11, due Wed April 20.    

Homework 12, due Sun May 1.    

Notes by Robert Friedman: Galois Theory I   II   III  IV
Part III contains the main theorem of Galois theory and examples of fields/subgroups correspondence.
Part IV is the Fundamental Theorem of Algebra and solving equations of degrees 3 and 4 in radicals.
These topics in Rotman: Solvability by Radicals (pages 71-75), Main theorems and their applications (pages 76-89).

My lecture notes: cyclotomic extensions insolvability of the quintic,
symmetric functions discriminants and degree 3 extensions

Review notes for the final: Part I Part II.

Solvable groups: Rotman Appendix B, pages 118-128.

Optional reading: Morandi, cyclotomic extensions.

Additional textbooks: There are many excellent textbooks that cover similar material, including