Math 404A: Introduction to Modern Algebra

Spring 2021

Lectures: MWF 10:30-11:20 on Zoom
Instructor: Jarod Alper (jarod@uw.edu)
Office hours: Wed 3-4 pm on zoom

TAs:

Paige Helms (phelms@uw.edu)
Adam Kapilow (akapilow@uw.edu)
- Office hours on zoom Thursday 11:30 am - 12:30 pm

Textbook:

Thomas Hungerford, Abstract Algebra: An Introduction, 3rd edition, Brooks/Cole, 2013.

Course content: The course will cover Galois theory, one of the crown jewels of modern mathematics. Galois’s mathematical approach in the early 19th century was strongly influenced by Abel, Bernoulli, Euler, Gauss, Legendre, Lagrange, Newton and many others stretching back to antiquity. Galois's ideas exerted an extraordinary influence on the course of modern mathematics, shifting emphasis away from explicit, messy algebraic calculations, and towards an understanding of the abstract structures that lay behind them. Using the techniques developed in this course, we will solve problems which had been open for thousands of years!

Schedule: A very brief summary of the content of each lecture will be posted here. The sections of the textbook quoted below are only a rough approximation of what we actually covered in class.

Date	Topics covered	Due	Video, slides and notes
Week 1
Mon Mar 29	Introduction: course summary and policies
Wed Mar 31	Cubic equations	Reflection 1	• video, slides • notes on cubics (Smyth)
Fri Apr 2	Cubic equations continued	• HW 1 • HW1 solns	• video, slides • Quick history of polynomials (T. Bösel) • Biography of del Ferro (St. Andrews collection) • Cubic dispute: L. Liberti, St. Andrews collection.
Week 2
Mon Apr 5	Review of group actions, structure of roots of polynomials in k[x], symmetric polynomials		• video, slides • Chapter 14--Group Actions (Judson) • notes on roots and symmetric polynomials (Smyth)
Wed Apr 7	Elementary symmetric functions, Fundamental Theorem on Symmetric Functions, Lagrange's solution to the quartic	Reflection 2	• video, slides • Lagrange's quartic solution (Smyth)
Fri Apr 9	Recap Lagrange's solution to quartic, review of groups, rings & fields	• HW 2 • HW 2 solns	• video, slides
Week 3
Mon Apr 12	Review of vector spaces and basis, field extensions (degrees and simple field extensions)	Quiz 1	• video, slides • Hungerford 11.1-11.2
Wed Apr 14	Simple field extensions, algebraic elements, minimal polynomials	Reflection 3	• video, slides • Hungerford 11.2-11.3
Fri Apr 16	Algebraic elements, algebraic field extensions, degrees of field extensions in towers	• HW 3 • HW 3 solns	• video, slides • Hungerford 11.1-11.3
Week 4
Mon Apr 19	Finite, algebraic and transcendental field extensions; sums and products of algebraic elements are algebraic	Quiz 2	• video, slides • Hungerford 11.3
Wed Apr 21	Ruler and compass constructions I	Reflection 4	• video, slides • Hungerford 15
Fri Apr 23	Ruler and compass constructions II	• HW 4 • HW 4 solns	• video, slides • Hungerford 15
Week 5
Mon Apr 26	Splitting fields	Quiz 3	• video, slides • Hungerford 11.4
Wed Apr 28	Uniqueness of splitting fields and discussion	Reflection 5	• video, slides • Hungerford 11.4
Fri Apr 30	Normal field extensions, splitting fields are normal, definition of separable field extensions	• HW 5 • HW 5 solns	• video, slides • Hungerford 11.4-11.5
Week 6
Mon May 3	Separable field extensions	Quiz 4	• video, slides • Hungerford 11.5
Wed May 5	Finite fields	Reflection 6	• video, slides • Hungerford 11.6
Fri May 7	Revisiting normal and separable field extensions; further discussion of finite fields	• HW 6 • HW 6 solns	• video, slides • Hungerford 11.4-11.6
Week 7
Mon May 10	First properties of the Galois group	Quiz 5	• Galois bio (MacTutor) • Mirzakhani bio (MacTutor) • The Beautiful Mathematical Explorations of Maryam Mirzakhani (Quanta magazine) • video, slides • Hungerford 12.1
Wed May 12	Examples of Galois groups and discussion	Reflection 7	• video, slides • Hungerford 12.1
Fri May 14	More on Galois groups, statement of the Fundamental Theorem of Galois Theory	• HW7 • HW 7 solns	• video, slides • Hungerford 12.2
Week 8
Mon May 17	Fundamental Theorem of Galois Theory	Quiz 6	• video, slides • Hungerford 12.2
Wed May 19	Discussion of Galois theory and homework	Reflection 8	• video, slides • Hungerford 12.2
Fri May 21	Wrapping up the Fundamental Theorem of Galois Theory: For K -> E-> L, E is normal over K <=> Gal(L/E) is normal in Gal(L/K)	• HW8 • HW8 solns	• video, slides • Hungerford 12.2
Week 9
Mon May 24	Galois's criterion	Quiz 7	• video, slides • Hungerford 12.3
Wed May 26	Discussion	Reflection 9	• video, slides
Fri May 28	Galois's criterion	• HW9 • HW9 solns	• video, slides • Hungerford 12.3
Week 10
Mon May 31	No class--Memorial Day
Wed Jun 2	Wrapping up Galois's criterion		• video, slides • Hungerford 12.3
Fri Jun 4	Discussion	• HW10	not recorded

Homework assignments: There will be 10 weekly homework assignments each submitted on Canvas. Homeworks will be due by 11 pm each Friday. The homework assignments constitute a large component of the class. Homework provides you an opportunity to engage directly with the course material and reinforce ideas from the lectures and textbook.

A selection of the homework problems will be graded and returned to you. Your homework average contributes to 55% of your final grade.

You should expect to spend a considerable amount of time working on the homework. Typically you may want to spend at least six hours outside of class on each homework.

The lowest homework score will be dropped. Homework extensions will not be granted except for extraordinary circumstances.

Collaboration: You are strongly encouraged to work collaboratively with fellow students on the homework assignments. Discussing problems and ideas for solutions with your classmates is one of the best ways to learn the material. You'll get the most benefit from working with others if you've already made a concentrated effort on your own first.

However, when writing up solutions, you must write the solutions in your own words. It is not acceptable to copy solutions written by another student or found online.
Writing: It is difficult to overstate the importance of being able to communicate your ideas and solutions--this is not only true in mathematics and generally schoolwork but in nearly every professional field. Homework is your chance to practice communicating mathematical ideas by writing clear and detailed solutions. To reemphasize this point, it is not enough simply to solve the problem; you must effectively communicate your solution. In some cases, once you think you've solved the problem, you may need to spend an equal amount of time on figuring out an effective way to communicate your solution.

Here are some tips on writing your solutions:
- Stating the problem: Begin each homework exercise by restating the problem itself--either in your own words or simply by copying the exercise verbatim.
- Writing mathematics is writing! Write full sentences. Every mathematical equation or expression should be accompanied with a full sentence explanation.
- Citing results: You may freely cite any results from lecture, earlier exercises, or statements from the textbook that have already been covered. If you do use a previous result, be sure to identify it clearly by stating its name, theorem number, or by repeating its statement.
- Typesetting vs handwriting: You may choose to submit either typeset or handwritten solutions. I highly recommend using LaTeX as it is the de facto standard in mathematics. If you handwrite your solutions, it is to your advantage to write clearly and legibly as otherwise the TAs may not be able to understand your solution.
- Writing several drafts: In some cases, the first time you writeup a solution it may be messy and poorly organized (especially if you are writing it by hand). Often in the process of writing your solution, you will realize there is an easier approach or you'll come up with cleaner notation. Don't hesitate to discard your first (or second...) attempt as a draft and start again.
- Submit your solutions in order: Please submit your solutions in the same numerical order as stated on the exercise. This will make it much easier on the TAs to grade.

Understanding: The ultimate goal should not be to merely finish the homework to get a good grade but rather to understand the problem and solution. A good gauge of a solid understanding is whether you can effectively communicate your solution to a classmate. After solving an exercise, you should sometimes ask yourself why exactly your approach worked. Will the same approach work on similar problems? Is there perhaps a simpler or alternative approach? Thinking about these types of questions will not only improve your understanding but will greatly assist in solving related problems.

Additional resources:
- How to read math textbooks by Katherine Stange.
- How learning math is similar to learning a foreign language by Katherine Stange.
- Homework strategies by Katherine Stange.
- Solving mathematical problems by Terence Tao.

Reflections: Self-reflections can serve as important educational devices. You are required to submit through Canvas a short self-reflection journal entry documenting your mathematical journey over the last week. Self-reflections are due 11 pm each Wednesday. These entries should be at least one or two paragraphs but you are welcome to write more. Reflections will be graded based on completion with a score of either 0 or 1. Your are allowed to miss one self-reflection.

The content of each self-reflection is up to you but here are some ideas for topics you may want to address:

What was your favorite result over the last week from this class?
What was the most challenging topic? What are your learning strategies to overcome this challenge?
Describe a mistake you made and what you learned from it.
Name two things you’re proud of yourself for this week, and one thing you might do differently next week.
Do you notice any connections between this material and other mathematical subjects you've studied?
What other topics are you curious about and want to learn more?
What learning strategies do you think might help in furthering your understanding of the material?

Quizzes: There will be 5-8 quizzes during the course each administered during your own time on Canvas. Each quiz is designed to take 20 minutes but you will be given a total of 30 minutes to compensate for the additional time needed to download/upload the quiz. The lowest quiz score will be dropped.

The content of each quiz will be directly based on the previous one or two homework assignments.

Course policies:

Grading:
- Homework: 55%
- Reflections: 5%
- Quizzes: 40%
The lowest homework score and quiz score will be dropped. This policy applies to students in good standing; students who engage in misconduct may have their assignments handled differently. Reflections are graded based only on completion and you are allowed to miss one.

To earn a 2.0 in the class, you need to earn at least a 50% final average.

Religious accommodation: Washington state law requires that UW develop a policy for accommodation of student absences or significant hardship due to reasons of faith or conscience, or for organized religious activities. The UW’s Reglious Accommodation Policy, including more information about how to request an accommodation, is available here. Accommodations must be requested within the first two weeks of this course using the Religious Accommodations Request form.

Student disability resources: UW offers disability accommodations. If you have already established accommodations with Disability Resources for Students (DRS), please communicate your approved accommodations to me at your earliest convenience so we can discuss your needs in this course. If you have not yet established services through DRS, but have a temporary health condition or permanent disability that requires accommodations (conditions include but not limited to; mental health, attention-related, learning, vision, hearing, physical or health impacts), you are welcome to contact DRS at 206-543-8924 or uwdrs@uw.edu or disability.uw.edu. DRS offers resources and coordinates reasonable accommodations for students with disabilities and/or temporary health conditions.

Student conduct code: The University of Washington Student Conduct Code (WAC 478-121) defines prohibited academic and behavioral conduct and describes how the University holds students accountable as they pursue their academic goals. Allegations of misconduct by students may be referred to the appropriate campus office for investigation and resolution. More information can be found online here.