Spring 2023
General Information
Syllabus
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 | 1/23 - 1/27 | Rings and modules. Examples: vector spaces, abelian groups, modules over F[x]. Submodules and quotients. Kernel, image, isomorphism theorem | HW1 | Feb 2, 2023 | |
| 2 | 1/30 - 2/3 | Generators and relations. Beginning of classification theory for abelian groups: row and column operations for integer matrices. | HW2 | Feb 9, 2023 | |
| 3 | 2/6 - 2/10 | Structure theorem for finitely generated modules over Z (aka abelian groups). Modules over PIDs | HW3 | Feb 16,2023 | |
| 4 | 2/13 - 2/17 | Structure theorem for modules over a PID. Jordan canonical form. | No HW this week | ||
| 5 | 2/20 - 2/24 | Group representations: basic definitions, irreducible representations, Schur's lemma. | HW4 | March 2, 2023 | |
| 6 | 2/27 - 3/3 | Group representations: complete reducibility, characters | HW5 | March 9, 2023 | |
| 7 | 3/6 - 3/10 | Fields and extensions. Finite and algebraic extensions. | HW6 | March 23, 2023 | |
| 3/13 - 3/17 | Spring Break | ||||
| 8 | 3/20 - 3/24 | Splitting field of a polynomial. Algebraic closure. Finite fields | HW7 | March 30, 2023 | |
| 9 | 3/27 - 3/31 | Examples of splitting fields. Automorphism group Aut(L/K). | HW8 | April 6, 2023 | |
| 10 | 4/3-4/7 | Galois extensions. Fundamental theorem of Galois theory | HW9 | April 13, 2023 | |
| 11 | 4/10-4/14 | Examples of Galois groups and corresponding subfields. Ruler and compass constructions | HW10 | April 20, 2023 | |
| 12 | 4/17-4/21 | Galois groups of polynomials. Symmetric polynomials. | No HW | ||
| 13 | 4/24 - 4/28 | Solvability in radicals. Solvable groups | HW11 | May 4, 2023 | |
| 14 | 5/1 - 5/5 |