The course will be primarily about groups and rings. In groups theory we will cover: the definition of a group, examples of groups like cyclic groups and permutation groups, subgroups, group homomorphisms, cosets of subgroups, Lagrange's theorem, group homomorphisms, normal subgroups, quotient groups and the isomorphism theorems. In ring theory, we will cover: the definition of a ring, ideals, quotients by ideals, ring homomorphisms, prime ideals, maximal ideals and fields. At the beginning of the class, we will cover some topics from elementary number theory such as the Euclidean algorithm, the fundamental theorem of arithmetic and modular arithmetic. If time permits, we will prove the classification of finite abelian groups and Sylow's theorems.
After taking the class, the students should know enough about groups and rings be able to follow a course on Galois theory. This is the theory of field extensions invented by Evariste Galois to prove that general quintic polynomial equations cannot be solved in radicals.
There will be two midterm held in class on dates which will be tentatively as follows.
| Midterm 1 | March 5. |
|---|---|
| Midterm 2 | April 16. |
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.
These are very rough notes which I have kept so that I remember approximately what has been said in class.