Math 81/111 Abstract Algebra

Term: Winter 2024

Evariste Galois
Lecture 01 (10464/10832)
Inst : Prof. Asher Auel
asher * auel AT dartmouth * edu
Time : Mon Wed Fri 10:10 - 11:15 am
X-hour Thu 12:15 - 1:05 pm
Loct : Kemeny 307
Office : Kemeny 339
Phone : No office phone!
Wed 1:00 - 2:30 pm
Thu 12:15 - 1:05 pm
Text : Abstract Algebra, 3rd Edition
David S. Dummit and Richard M. Foote
John Wiley & Sons.
ISBN-13: 978-0-471-43334-7.
 Course syllabus and homework schedule.

Description of course: The main object of study in field theory and Galois theory are the roots of single variable polynomials. Many ancient civilizations (Babylonian, Egyptian, Greek, Chinese, Indian, Persian) knew about the importance of solving quadratic equations. Today, most middle schoolers memorize the "quadratic formula" by heart. While various incomplete methods for solving cubic equations were developed in the ancient world, a general "cubic formula" (as well as a "quartic formula") was not known until the 16th century Italian school. In the late 18th and early 19th century, it was conjectured by Gauss, and nearly proven by Ruffini, and then finally by Abel, that the roots of the general quintic polynomial could not be solvable in terms of nested roots. Galois theory, introduced by √Čvariste Galois the 1830s, provides a satisfactory explanation for this, as well as to the unsolvability (proved independently in the 19th century) of several classical problems concerning compass and straight-edge constructions (e.g., trisecting the angle, doubling the cube, squaring the circle). More generally, Galois theory is all about symmetries of the roots of polynomials. An essential concept is the field extension generated by the roots of a polynomial. The philosophy of Galois theory has also impacted other branches of higher mathematics (Lie groups, topology, number theory, algebraic geometry, differential equations).

This course will provide a rigorous proof-based modern treatment of the main results of field theory and Galois theory. The main topics covered will be irreducibility of polynomials, Gauss's lemma, field extensions, minimal polynomials, separability, field automorphisms, Galois groups and correspondence, constructions with ruler and straight-edge, theory of finite fields. Some advanced topics, such as infinite Galois theory and Galois cohomology, will be included. The grading in Math 81/111 is very focused on precision and correct details. Problem sets will consist of a mix of computational and proof-based problems.

Expected background: Previous exposure to linear and abstract algebra (Math 24 and Math 71) is required. If you have had Math 22 and/or Math 31, please consult with the instructor about enrolling in the course.

Homework 40%
Takehome midterm exam   25%
Final exam (08 Mar) 35%
Grades: Your final grade will be based on weekly homework, a takehome midterm exam, and a final in-class exam. While significant emphasis is placed on exams, completing your weekly homework will be crucial to your success on the exams and in the course.
Group work, honestly: Working with other people on mathematics is highly encouraged and fun. You may work with anyone (e.g., other students in the course, not in the course, tutors, ...) on your homework problems. If done right, you'll learn the material better and more efficiently working in groups. The golden rule is:
Work with anyone on solving your homework problems,
but write up your final draft by yourself.
Writing up the final draft is as important a process as figuring out the problems on scratch paper with your friends, see the guidelines below. If you work with people on a particular assignment, you must list your collaborators on the top of the first page. This makes the process fun, transparent, and honest.

External resources: Mathematical writing is very idiosyncratic; if your proofs are copied, it is easy to tell. You will not learn (nor adhere to the Honor Principle) by copying solutions from others, or from external sources such as internet forums and generative artificial intelligence (GAI) output.

Concerning internet forums (e.g., math.stackexchange), you are free to look at them and use any understanding you've gained from them in your course work, of course, subject to the above rules. Just be warned that these forums often contain incorrect or circuitous solutions, misleading discussions, use of techniques outside of the course material, and other material that may be detrimental to your learning process. Even the time that it takes to repeatedly search for solutions and read through dozens of forum posts could be better spent learning the material on your own or composing a question via email to the instructor.

Concerning GAI (e.g., ChatGPT), you are free to experiment with asking questions, but be warned that these systems are currently still very bad at deductive reasoning, and that the output may contain a mix of correct, incorrect, and unverified statements. Ask them to prove something false, they will work hard to do so, often giving contradictory answers. Therefore, I would be very careful with using these tools as learning resources on your own.

Attendance: You are expected to attend class, including required X-hour sessions, in person unless you have made alternative arrangements due to illness, medical reasons, or the need to isolate due to COVID-19. For the health and safety of our class community, please follow Dartmouth's health guidance.

Accommodations: Students requesting disability-related accommodations and services for this course are required to register with Student Accessibility Services and to request that an accommodation email be sent to me in advance of the need for an accommodation. Then, students should follow-up with me to determine relevant details such as what role SAS or its Testing Center may play in accommodation implementation. This process works best for everyone when completed as early in the term as possible. If students have questions about whether they are eligible for accommodations or have concerns about the implementation of their accommodations, they should contact the SAS office. All inquiries and discussions will remain confidential.

Additional notes

X-hour: Generally, the X-hour will alternate between problem solving sessions and mini review courses on linear algebra and abstract algebra conducted by TA Lucy Knight. Otherwise, it will revert to office hours or midterm review, and occasionally will serve another purpose (e.g., extra or make-up lecture time) that will be announced in advance.

Homework: Weekly homework will be due on Friday by 5 pm on Canvas. Each assignment will be posted on the syllabus page the week before it's due.

Homework might contain advanced problems, which will introduce additional topics or involve more difficult proofs. These are suggested for everyone, but are only required for graduate students.

If you know in advance that you will be unable to submit your homework on time, you must make special arrangements in advance.

You might consider taking the opportunity to learn LaTeX. Otherwise, you can write out your solutions, neatly and straight across the page, on clean paper, with nice margins, scan them, and upload them.

Your lowest homework score above 50% from the semester will be dropped.

Exams: The takehome midterm exam will be assigned over a week in February. You will not be able to work together during the take-home midterm exam.

The final exam will take place 11:30 am - 02:30 pm on Friday 08 March. The use of electronic devices of any kind during the final exam will be strictly forbidden.

Homework guidelines: Generally, a homework problem in any math course will consist of two parts: the creative part and the write-up.

  • The creative part: This is when you "solve" the problem. You stare at it, poke at it, and work on it until you understand what's being asked, and then try different ideas until you find something that works. This part is fun to do with your friends; you can do it on the back of a napkin. If you're having trouble, even in understanding what the problem's asking, use the resources available to you: my office hours, teaching assistants' office hours, weekly tutoring sessions, etc. Ask for help as early as you can! This part should all be done on "scratch paper."

  • The write-up: Now that everything about the problem is clear in your mind, you go off by yourself and write up a coherent, succinct, and nicely written solution on clean sheets of paper. Consider this your final draft, just as in any other course. This part you should definitely NOT do with your friends.