Math 370 Fields and Galois Theory

Semester: Spring 2019

Evariste Galois
Lecture 01 (22059)
Inst : Prof. Asher Auel
asher * auel AT yale * edu
Time : Tue Thu 11:35 am-12:50 pm
Loct : LOM 215
Office : LOM 210
Phone : (203) 432-4187
Office
hours :
Tue 1:45 - 3:45 pm
Text : Fields and Galois Theory (v4.60)
J. S. Milne
Available at www.jmilne.org/math/
 Course syllabus and homework schedule.
 Office hours during reading period :
  Wed, May 1, 12:00 - 1:00 pm
 Final Review Sessions :
  Wed, May 1, 02:45 pm - 05:15 pm LOM 215 (Auel)
  Thu, May 2, 12:30 pm - 02:30 pm Common Room (Arthur)  
Final exam review guide
Peer Tutor
Peer tutor : Arthur Azvolinsky
Office hours : Monday and Wednesday 6:00 - 8:00 pm
Loct : DL 431/Math Dept Common Room

Description of course: The main object of study in 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. The grading in Math 370 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 is required. For example, the contents of Math 225 Linear Algebra and Matrix Theory and Math 350 Abstract Algebra, are recommended.

Homework 20%
Midterm exam 1   15%/25%
Midterm exam 2   15%/25%
Final exam (03 May) 30%/40%
Grades: Your final grade will be based on weekly homework, two midterm exams, and a final exam (see formula below). While more overall emphasis is placed on exams than on homework assignments, 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. Mathematical writing is very idiosyncratic - we will be able to tell if papers have been copied - just don't do it! You will not learn by copying solutions from others or from the internet! Also, 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.



Policies

(or otherwise the small print)

Homework: Weekly homework will be due in-class on Thursday. Each assignment will be posted on the syllabus page the week before it's due.

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. Late homework, without a dean's excuse, will be accepted up to one week after the deadline and will be worth only 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.

Do not copy homework solutions from internet resources!

No homework will be due during the week of the midterm exam.

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

Exams/quizzes: The midterm exams will take place in-class on Tuesday 19 February and Tuesday 09 April. The final exam will take place 02:00 - 05:30 pm on Friday 03 May.

Your total score will be calculated as the maximum of:
20% Homework + 25% Midterm 1 + 25% Midterm 2 + 30% Final
20% Homework + 15% Midterm 1 + 25% Midterm 2 + 40% Final
20% Homework + 25% Midterm 1 + 15% Midterm 2 + 40% Final

Make-up exams will only be allowed with a dean's excuse.

The use of electronic devices of any kind during quizzes and exams is 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.