This course is primarily about field theory and Galois' theory of equations. This theory associates a group, called the Galois group, to a polynomial equation. Properties of the group translate into important properties of the equation. For example, using Galois theory we will show that the equation x 5 - x -1 cannot be solved using radicals.
I plan to follow Milne's text fairly closely augmenting it when necessary with review. To warm up, we will begin by proving the fundamental theorem of finite abelian groups following Milne's group theory book. Then, after a short review of linear algebra and polynomials, we will follow Milne in discussing field extensions of finite degree. As an application, I will show that it is impossible to trisect a general angle.
After that I will introduce the notions of stem fields, splitting fields, Galois extensions and Galois groups. I will then prove the Fundamental Theorem of Galois theory, which sets up a correspondance between subfields of a Galois field and subgroups of its Galois group. At this point we will compute several simple examples of Galois groups to get a feel for what they are.
We will then cover two topics in pure Group theory: Sylow's theorem and the theory of solvable groups. For these topics I will loosely follow Milne's notes on Group Theory which are also freely available on the web.
I will then discuss Kummer extensions and unsolvability of extensions by radicals. In terms of Chapters in Milne's books, my plan is to cover Chapters 1-5 of Field Theory, and Chapter 3 of Group Theory. I will also cover the sections of Group Theory pertaining to finite abelian groups and to solvable groups. Those are the section on commutative groups in Chapter 1 and the first two sections of Chapter 6. If time permits I will cover Chapter 8 of Field Theory, which discusses an extension of Galois' theory due to Grothendieck.
There will be two midterms held in class on dates which will be tentatively as follows.
| Midterm 1 | March 4. |
|---|---|
| Midterm 2 | April 17. |
If you miss an exam due to illness or other unavoidable circumstances, I will either give you a makeup exam or replace the missed exam grade with the unmissed exam. Which one I do depends on how many people miss the exam.
Course grades will be computed as follows.
| Homework | 20% |
|---|---|
| Midterms | 40% |
| Final | 40% |
Homework will be assigned weekly and due on Wednesday's. It will be assigned on the homework web page linked here and at the top of this syllabus web page. Late homework will not be accepted but the lowest two homework grades will be dropped.
I want every student to feel free to questions in class. Without student questions, even ones that might sometimes seem silly, lectures can become very dry and monotonous. So questions and comments are more than welcome!
On the other hand, please be courteous to other students by paying attention to the lecture and not carrying on conversations with each other. These kind of private conversations can be distracting both to me and everyone else.
Please make sure that your cell phone does not ring during the lecture, and please do not use headphones. Please do not use laptops for anything not related to the class, and, if you do bring a laptop, please sit in the back so that your screen is not distracting to others.