Galois Theory (Math 5658), Spring 2014
Instructor: Andrew Obus
email: obus [at] virginia.edu
office: Kerchof Hall 208
MW 2:00-3:15, Clark Hall G004. Please ask questions if anything in lecture is unclear.
Lectures will run the entire 75 minutes.
Please show up on time!
Galois Theory, 2nd ed.,
by David A. Cox. Available at the bookstore and on amazon.
There is an electronic copy on reserve in the
library. This is a large book, but a lot of that is due to having
very detailed exposition, along with copious historical and
mathematical notes.Other references you might enjoy include:
- Galois Theory by
Emil Artin (the original lecture series on Galois theory from a modern
perspective. Takes an approach that emphasizes linear algebra
more than Cox's book does. From 1942, but very clear and
readable!) Available here.
- Algebra by Michael Artin (Emil's son! The last two chapters give a more concise account of Galois Theory than Cox does.)
- Galois Theory, by Ian Stewart (roughly follows E. Artin's presentation, but includes a great chapter on history.)
- Galois Theory for Beginners, by J÷rg Bewersdorff (informal, requires less background, but makes you get your hands dirty with polynomials.)
- Galois Theory,
by Steven H. Weintraub (more advanced, goes very quickly through the
basic theory, although techinically does not require more background
than Cox's book).
theory is a fascinating amalgam of group theory and field theory.
It has its origins in studying symmetries of roots of polynomials (for
instance, complex conjugation exchanging roots of real quadratic
polynomials with negative discriminant), and now involes studying the
symmetries (or "automorphisms") of abstract fields. Galois theory
explains why most rational polynomials of degree ≥ 5 defined over the
rational numbers do not have solutions expressible by radicals, why all
rational polynomials of degree < 5 do have solutions expressible by radicals (i.e., why there is
a quadratic, cubic, and quartic formula), why the regular 17-gon can be
constructed with ruler and compass, and much more. We will spend
much of the course building toward the Fundamental Theorem of Galois Theory,
which gives a correspondence between subfields of certain field
extensions and subgroups of a group called the "Galois group of the
extension." This so-called "Galois correspondence" not only has many
applications, but also is a prototype for many similar-looking theorems
In particular, I hope to cover most of Chapters 1-8 of Cox's book
in the first 11 or so weeks of class. This is the core material
of the course. There are many topics we can discuss for the last
couple of weeks (explicit solutions of quartics and quintics, ruler and
compass/origami constructions, the inverse Galois problem, Galois
theory in characteristic p, transcendental extensions...). I will
seek the class's input!
╔variste Galois, the inventor of Galois theory, also had an amazing
life story, culminating in his death at age 20 in a duel. I will
spend some time discussing the history of Galois himself, as well as
the history of his mathematics.
Mondays 1-2, Fridays 12-1. Kerchof Hall 208 (my office).
If these times do not work for you, please make an appointment with me.
assignment will consist of two parts. The first part will consist
of a few basic problems, which are not to be handed in. These are
meant to check your understanding --- if you have questions about them,
I am happy to help! The second part will have 3-4 more involved
problems (usually proofs) which I will collect and grade. Please
make homework as neat as possible, and please make your arguments as
clear as you can (be honest when you don't know how to do something)!
Homework will be graded in a somewhat non-traditional way. Each
problem will get a grade of "Correct" or "Redo." "Correct" means
that there are only very minor problems with the argument, which I can
point out quickly. "Redo" means that there is a reasonably
serious gap in your reasoning, or that the argument does not make sense
(I will try to write some comments in this case). Problems marked
"Redo" can be handed in within a week of being returned for 80%
credit. If a problem handed in a second time gets a "Redo," then
it can be handed in a third time for 60% credit, a fourth time for
40% credit, etc. If you elect not to redo a problem, then it
earns 0% credit. My feeling is that this allows for better
learning than standard homework procedure, and is workable for a small
Homework will be assigned more or less weekly, and will be posted on Collab. Grades will be posted there as well.
Exams There will be an in-class
midterm on October 27th. There will be a take-home final exam
given out on December 1st, in lieu of a regular homework
assignment. It will be due either December 5th or December 7th,
depending on the preference of the class!
Your grade will be calculated as follows: 10% Participation, 20%
Midterm, 30% Homework, 40% Take-Home Final. This is a small
class, so it does not make sense to have a strict curve. If you
all show a good understanding of the material, you can all earn high
allows anonymous comments to be submitted to instructors. Please
take advantage of this if there is something you want to say!