Syllabus and Homeworks
There will be weekly homeworks, posted on Th. The homeworks will be announced on the course web page.
The syllabus below is tentative and subject to change.
Week |
Dates |
Notes |
Topics |
HW |
Due date |
1 |
1/28 - 2/1 |
|
Rings and modules. Examples: vector spaces, abelian groups, modules over F[x]. Submodules and quotients |
no HW |
|
2 |
2/4 - 2/8 |
|
Kernel, image, isomorphism theorem. Generators and relations. |
HW1 |
Th, 2/14 |
3 |
2/11 - 2/15 |
Class on Tu was cancelled. |
Structure theorem
for finitely generated modules over Z (aka abelian groups). |
HW2 |
Th, 2/21 |
4 |
2/18 - 2/22 |
|
Structure theorem
for finitely generated abelian groups - completion: bringing an integer matrix to a canonical form. Principal ideal domains. |
HW3 |
Th, 2/28 |
5 |
2/25 - 3/1 |
|
Structure theorem for modules over a PID. Jordan canonical form. |
HW4 |
Th, 3/7 |
6 |
3/4 - 3/8 |
|
Group representations: basic definitions, irreducible representations, Schur's lemma. |
HW5 |
Th, 3/14 |
7 |
3/11 - 3/15 |
|
Group representations: complete reducibility, characters |
No HW |
|
|
3/18 - 3/22 |
Spring Break |
|
|
|
8 |
3/25 - 3/29 |
|
Fields and extensions. Splitting field of a polynomial |
HW6 |
Th, 4/4 |
9 |
4/1 - 4/5 |
|
Finite extensions; degree. Algebraic extensions. |
No HW |
|
10 |
4/8-4/12 |
|
Algebraic extensions continued. Constructions with ruler and compass |
HW7 |
Th, 4/18 |
11 |
4/15-4/19 |
|
Cyclotomic polynomials; constructions of regular n-gon. Normal extensions. |
HW8 |
Th, 4/25 |
12 |
4/22-4/26 |
|
Galois group. Main theorem of Galois theory |
HW9 |
Th, 5/2 |
13 |
4/29 - 5/3 |
|
Examples of Galois group. Symmetric polynomials |
HW10 |
Th, 5/9 |
|