Math 81/111 Abstract Algebra

Term: Winter 2020

Evariste Galois
Lecture 01 (10963)
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 343
Office : Kemeny 339 240
Phone : No office phone yet!
Mon 2:00 - 4:00 pm
Thu 12:15 - 01:15 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. 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 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 me about enrolling in the course.

Homework 40%
Takehome midterm exam   25%
Final exam (09 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. 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 the internet.

Additional notes

X-hour: The X-hour will usually consist of office hours. On the occasional week, the X-hour will serve another purpose (e.g., extra or make-up lecture time) and I'll announce it in advance.

Homework: Weekly homework will be due at the beginning of class each Friday. 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 at the correct time and place, you must make special arrangements ahead of time. Otherwise, late homework will be accepted up to one week after the deadline and will be worth 50%.

Your homework must be stapled (or otherwise securely fastened) together, with your name clearly written on the top. Consider the pieces of paper you turn in as a final copy: written neatly and straight across the page, on clean paper, with nice margins and lots of space, and well organized. You might consider taking the opportunity to learn LaTeX.

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 08:00 am - 12:00 pm on Thursday 12 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.