Program Overview

Math 81/111 Abstract Algebra

Course Description

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 the instructor about enrolling in the course.

Course Structure

• Homework: 40%
• Takehome midterm exam: 25%
• Final exam: 35% Grades will be based on weekly homework, a takehome midterm exam, and a final in-class exam. While significant emphasis is placed on exams, completing weekly homework will be crucial to success on the exams and in the course.

Homework Guidelines

• Homework problems 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 friends; you can do it on the back of a napkin.
  • 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 friends.
• 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.
• 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 on Thursday, 12 March, from 08:00 am - 12:00 pm. The use of electronic devices of any kind during the final exam will be strictly forbidden.

Group Work and Honesty

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 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. 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.