Course format

Online.

If you are taking this class, but have any concerns about accessibility--including to fast-enough internet for online lectures, to technology (such as a laptop), et cetera--let Hiro know as soon as possible.

Class Meetings: TuTh 5 - 6:20 PM.

Office Hours: Tue 2:30 PM - 4:30 PM.

Course description

Definitions and elementary properties of groups, rings, integral domains, fields and vector spaces with great emphasis on the rings of integers, rational numbers, complex numbers, polynomials, and the interplay between algebra and geometry.

Beyond a fluency with the above topics, another goal of this class is for you to become familiar with mathematical thinking---questioning and understanding why definitions exist, identifying when you or another communicator is being precise or imprecise (and for what purpose), developing tastes that are rooted in practice and informed experience, exploring the mathematical landscape on your own.

Textbook and resources.

You do not need to buy a textbook for this course. The following are freely available resources:

1. Judson's free text
2. Milne's various notes
3. Beachy's various notes
4. Conrad's notes on various topics
5. Dick Gross's lectures

The syllabus

Here is the course syllabus as of January 17.

Collaboration policy

I strongly encourage all of you to collaborate. Please do so. If you do, you must indicate clearly on every assignment that you have collaborated, and indicate with whom. However, write solutions on your own. It is fine to think through problems and find solutions with each other, but when it comes to the act of writing your homework, you must do so without assistance from another. This is because the act of solving something and writing a mathematical proof are two different skills, and I want you to also hone the latter. As an extreme anti-example, copying and pasting solutions/proofs will not be tolerated. To reiterate, you may not write solutions together.

Recordings of Zoom Lectures

Can be found here.

Notes

1. Tue, Jan 18. Operations. (Updated February 10.)
2. Thu, Jan 20. Axiomatic Thinking. Rings. (Updated January 21.)
3. Tue, Jan 25. Matrices and two-dimensional vectors. (Updated January 26.)
4. Thu, Jan 27. Acting on the Euclidean plane. (Updated January 30.)
5. Tue, Feb 1. Complex numbers.
   Thu, Feb 3. [Class cancelled due to cold weather per university.]
6. Tue, Feb 8. Ring homomorphisms. (Updated February 10th.)
7. Thu, Feb 10. Polynomial rings. (Updated Feb 11th.)
8. Tue, Feb 15. Algebraic sets. (Updated Feb 15th.)
9. Thu, Feb 17. Ideals and algebraic functions.
10. Tue, Feb 22. Quotient rings and ring isomorphisms. (Updated Feb 23rd.)
11. Thu, Feb 24. Exercise day. Hiro not in class. We will have a guest lecturer, Max Warshauer. Here are his notes.
12. Tue, Mar 1. Some computations, and polynomial division.
13. Thu, Mar 3. Review of rings, or covering ring topics we didn't have time to cover.
14. Tue, Mar 8. Symmetries and groups.
15. Thu, Mar 10. Examples of groups coming from rings. Basic group facts. Updated March 11.
    Spring Break!
16. Tue, Mar 22. Product groups and subgroups.
17. Thu, Mar 24. Groups actions and order.
18. Tue, Mar 29. Dihedral and symmetric groups.
19. Thu, Mar 31. Conjugation and conjugacy classes.
20. Tue, Apr 5. Orbit-stabilizer.
21. Thu, Apr 7.Lagrange's Theorem and the Class Equation.
22. Tue, Apr 12. Kernels and normal subgroups.
23. Thu, Apr 14. Quotient groups and first isomorphism theorem.
24. Tue, Apr 19. (Guest lecturer) Linear fractional transformations.
25. Thu, Apr 21. (Possible guest lecturer) Elliptic curves.
26. Tue, Apr 26. Symmetries of a cube.
27. Thu, Apr 28. (Last day of class!)
28. Tue, May 3. No class. (I incorrectly wrote that this would be the last day of class in a past rendition of this website.)