Math 172: Abstract Algebra II

Course meets: MW 11am, Shanahan 3465

Office Hours: Tuesdays 9am-10am and 3pm-4pm, or by appointment.

Grader: Ryan O’Dowd, ryan.o’dowd@cgu.edu

The main topics of this course will be field extensions and Galois Theory, and additional topics as time permits.

Required Text

Dummit and Foote, Abstract Algebra (3rd edition). Doing the reading will be essential for success in this course. Also as a second-semester course, I will expect a certain level of mathematical maturity. This means that sometimes I will not prove simple statements in class; you should get in the habit of working out some details for yourself, and doing the reading.

Prerequisite

Algebra I (Math 171)

Grading Policy

A midterm (25%), a final (25%) and a homeworks (25%) with the lowest homework score dropped, and the strongest component of the three will be worth 50%.

Honor Code

All are expected to abide by the HMC honor code. Though cooperation on homework assignments is encouraged, you are expected to write up all your solutions individually. It is appropriate to acknowledge the assistance of others. Part of the fun of this course is the struggle, as well as the joy of discovering the solution for yourself. Please note: using solutions found online or solutions of previous students will be regarded as a violation of the HMC Honor Code and will be handled accordingly. I encourage you instead to discuss the ideas with me or each other!

Homeworks

Homeworks, assigned weekly, turn in Wednesdays (in class) or by 4pm in the bin outside my office door. Homeworks will be announced on this webpage.

On Learning: Read this article by Carol Dweck.

On Writing Mathematics Well: Read my handout on good mathematical writing.

LaTeX homework sets, courtesy of Luis Ramirez, may be found here: (but you are responsible for checking they are correctly transcribed): https://github.com/magh-marshmallow/Galois_HM

REVIEW 7.1–7.4 and READ 7.5
(R means READ but you do not have to do the problem.)

DO 7.3 ( 1, 10, R19, 26, 28 ) 7.4 ( 10 ) and Problem A:

A. (part i) Let $R$ be an integral domain. As you conjectured in class, prove that the units in $R[x]$ are precisely the constant polynomials $p(x)=u$ where $u$ is a unit in $R$ .
(part ii) On the other hand, show that $p(x)=1+2x$ is a unit in $R[x]$ , where $R= \frac{\mathbb Z}{4 {\mathbb Z}}$ . ( $\mathbb Z$ = integers.)

SKIM CHAPTER 8, READ 9.1–9.3
DO 7.4 ( 19 ) 7.5 ( 3 ) 8.2 ( 2, 3 ) 9.1 ( 4 )[You will probably find 7.5.3 to be the most involved problem; so you may wish to do it last. For several of the others, I encourage you to think about modding out a ring by an ideal.]

READ 9.4–9.5
DO 9.1 ( R2, R6, 7 ) 9.2 ( 1, 2, 3, R4, R6 ) 9.3 ( 3, R4 )

READ 13.1–13.2
DO 9.4 ( 1bcd, 2bc, R5, 6ac, R7, 11, R20 ) 13.1 ( 1 )

READ 13.4
DO 13.1 ( 3, 5 ) 13.2 ( R1, 3, 4, 12, 14 )

READ 13.4–13.6
DO 13.3 ( 1, R2 ) 13.4 ( 2, 3, 4 ) and these problems:

Problem C. Find a real number $u$ such that $Q(\sqrt{3},\sqrt{5})=Q(u)$ .

Problem D. Suppose $K$ is an extension of $F$ , and $\phi: K -> K$ is an isomorphism that leaves every element of $F$ fixed.
Show that any polynomial in $F[x]$ that has a root $r$ in $K$ also has $\phi(r)$ as a root.

Since there is a midterm due Thursday 3/12, there is no HW this week. Also, no HW over Spring Break.

With the COVID-19 crisis and the disruption that it has caused and will continue to cause, here are a few changes I’m making to this course:

Principles
- A higher priority than this course is your health and the health of your loved ones.
- We must have flexibility while planning ahead for possible further disruption, including obstacles you may have to online access or focused study while at home, and accommodations in case you (or I) get sick from COVID-19.
- Focusing on the ideas of this course can be an important way for you to feel normal in working towards the future, in a time of great upheaval and uncertainty.
Following from these principles, here are changes to our original plan.
- You will have the option to take this course Pass/No Credit. If you are an HMC student, a Pass will count towards your degree requirement (even if it would not normally do so). If you are not an HMC student, you should check with your registrar to see what your school policy is for Pass/No Credit. You can exercise this option at any time just by sending me an e-mail—you do not have to petition the registrar.
- For the homework average, the lowest two homework scores will be dropped.
- Lectures will be held as usual during the normally scheduled time at 11am Pacific, but recordings will be available to watch afterwards.
- We will figure this out as we go and may have to improvise.

Try out the Course Zoom Room.

Assemble 3 questions you have about the course material.
Discuss with a group of 2-5 others in the Su Gathering Space (in Zoom).
Show up in the Su Gathering Space at TUESDAY 6-9pm Pacific time, or self-assemble at a different time. (Remember WEDNESDAYS are reserved for the other class.)
Try out the Whiteboard feature in Zoom, or move to goboard.com.
See Sakai/email for links to Su Gathering Space.

READ 14.1–14.2, 14.6
DO 14.1 ( 7 ) 14.2 ( 3, 13, 14 )

All homework will now be submitted through Gradescope. Access this through Sakai, on the left tab. You will need to upload by scanning your homework, here is some guidance on how to do that: https://gradescope-static-assets.s3-us-west-2.amazonaws.com/help/submitting_hw_guide.pdf. Follow these Gradescope instructions to tag which portion of your homework is associated with the four problems: https://youtu.be/KMPoby5g_nE

READ 14.6
DO 14.1 ( 1, 4 ) 14.2 ( 1, 11 ).

READ 14.6, 14.7
DO 14.7 ( 1 ) 14.6 ( 2ac, 22, 23 ).

SKIM 9.6, 15.1. The main ideas are here, but I’m supplementing what’s here with my lectures.

14.6 ( 4, 37 [read the text above the problem], 38, 46 )
In problem 46, E and F are unspecified fields (that you get to specify)–it might be helpful to know that any group is isomorphic to a subgroup of some symmetric group (Cayley’s Theorem).

This is your final homework! It does not depend on this week’s lectures.

All late homework must be turned in by May 3 at 4pm.

I will have FINAL EXAMS available by Monday afternoon May 4.

Francis Su

Benediktsson-Karwa Professor of Mathematics, Harvey Mudd College