Galois Theory (Math 5658), Spring 2014  

 
Instructor: Andrew Obus
email: obus [at] virginia.edu
office: Kerchof Hall 208
phone: 434-424-4930
website: http://people.virginia.edu/~aso9t/math5658f14.html

 

Lectures

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!



Textbook

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:

Content

Galois 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 mathematics.

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.


Office Hours

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.


Homework

Each homework 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 class.

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!

Grading

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 grades!

Comments

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