Math 103B Spring 2023 Section B

General Logistics

The textbook is free online, via the above link.

Syllabus PDF

PDF

Class Overview

This is the continuation of Math 103A. We will talk about ring theory (with a focus on commutative rings with identity) and field theory in this course. Topics include rings, homomorphism, ideals, quotient rings, isomorphism theorems, polynomial rings, integral domains, fields, finite fields, and Galois theory.

Attendance is not part of your grade. However, there will not be podcast or recordings available, so you are strongly suggested to come to lectures. If you have to miss a class, be sure to ask your classmates for what you have missed, and come to office hours to catch up if needed.

This class focuses on understanding the examples. However, there will still be a lot of theorems and proofs in this class, both in lecture and in your homework. Please review your proofs if you feel rusty.

You are expected to read the textbook before each class. You can skip all the proofs unless I tell you to read them. There is a canvas exercise on your reading due before each class (except on midterms).

Office Hour

My office hour: MWF 4-5pm at our classroom.

If you are not sure what "office hour" is, it is an opportunity for you to talk to me about the class, seek help in topics you are unsure about, discuss the homework problems you are stuck at. A lot of students tend to be intimidated to talk to the instructor, fearing that they are judged for not knowing the material. The fact is, we are trying to help you succeed, and we do not evaluate student at all during office hours. A student who comes to office hours regularly will generally leave a good impression to the instructor, in contrast to the misbelief that the instructor would think such student is not performing well. So, if you need anything, come to office hours. Seek help soon, rather than later!

Late Join Policy

If you wish to enroll in this class (e.g. you are currently on the waitlist), you should keep up with the class since day 1. In particular, you need to attend all lectures, submit all the homework on time and be present for progress checks.

(IMPORTANT) When you are finally officially enrolled to the class, please read the ''Welcome Message'' on Canvas Announcements, and complete everything it instructs you to do.

Homework

Homework is due weekly on Mondays at 11:59pm, starting on week 2. Please read the homework instructions before working on your first homework. The homework questions will be listed below.

• HW 1: Ch 14.5 Q2, 3b (i.e. corresponding to 2b), 4, 5, 6, 20. Ch 16.7 Q3, 17, 31, 33.
• HW 2: Ch 16.7 Q6, 7, 8, 11, 12, 16, 18, 27, 35, 37.
• HW 3: Ch 16.7 Q4abe, 13, 22, 25, 26, 39. Ch 17.5 Q1, 6, 18, 29.
• HW 4: Ch 17.5 Q3, 4, 7, 11, 12, 13, 17, 25, 26, 27. Solution.
• HW 5: Ch 17.5 Q8, 9, 15, 20, 21, 24. Ch 18.4 Q6, 8, 16. Solution.
• HW 6: Ch 18.4 Q7, 10, 12, 17, 20. Ch 21.5 Q2acfhi, 5, 10, 13, 15. Solution.
• HW 7: Ch 21.5 Q1, 4, 12, 14, 16, 21, 22, 26, 27, 28. Solution.
• HW 8: Ch 21.5 Q3, 9, 11, 17, 18, 20, 21, 23, 25. Ch 22.4 Q4. Solution.
• HW 9: Ch 22.4 Q2, 5, 8, 17, 18, 19, 20, 21, 23. Ch 23.5 Q2. Solution.
• Practice Problem for week 10: Ch 23.5 Q1, 3, 6, 7, 14, 18. Also:

  The extension Q(zeta_5) / Q has Galois group isomorphic to Z/4. A generator of the group is phi(zeta_5) = zeta_5^2. Compute the fixed field by phi^2.

  The extension Q(zeta_7) / Q has Galois group isomorphic to Z/6. A generator of the group is phi(zeta_7) = zeta_7^3. Compute the subgroup corresponding to F = Q(zeta_7 + zeta_7^2 + zeta_7^4). You should do so by computing all phi^k in the Galois group, and see how they act on the generator of the subfield. What is the degree of F/Q?

  For Q(zeta_7)/Q, can you find the subfield of degree 3 over Q?

Discussion Worksheet

• Week 1
• Week 2
• Week 3
• Week 5
• Week 6
• Week 8
• Week 9
• Week 10