MTH 461: Survey of Modern Algebra
University of Miami, Spring 2024
Instructor: Christopher Scaduto
Email: c.scaduto @ math.miami.edu
Office: Ungar 525
Office hours: 10-11 Tuesday, 12:15-1:15 Thursday, or by appointment
Time and Location: TuTh 11:00AM - 12:15PM,Cos Science 213
Course syllabus can be found here.
Writing assignment (for writing credit): info here.
Homework Assignments
All homeworks are worth the same, even if graded out of different totals.
| Assignment |
Due Date |
Remarks |
| Homework 1: pdf |
1/25/24 |
|
| Homework 2: pdf |
2/1/24 |
|
| Homework 3: pdf |
2/8/24 |
|
| Homework 4: pdf |
2/15/24 |
|
| Homework 5: pdf |
2/29/24 |
|
| Homework 6: pdf |
3/7/24 |
|
| Homework 7: pdf |
3/28/24 |
|
| Homework 8: pdf |
4/11/24 |
|
| Homework 9: pdf |
4/30/24 |
|
The following are solutions to some of the homework exercises.
Note that problems in these files may be slightly different from those in the homeworks.
solutions set 1
solutions set 2
solutions set 3
solutions set 4
solutions set 5
solutions set 6
solutions set 7
solutions set 8
Course Schedule
| Date |
Lecture Content |
Reading |
| 1/16/24 |
Syllabus overview, introduction, definition of a group |
note 1 |
| 1/18/24 |
Cayley tables, basic properties of groups, subgroups |
note 2 |
| 1/23/24 |
Integers mod n |
note 3, note 4 |
| 1/25/24 |
More integers mod n, orders of elements |
note 5 |
| 1/30/24 |
Symmetries of objects. Permutations. Symmetric groups. |
note 6, note 7 |
| 2/1/24 |
More symmetric groups. Alternating groups. |
note 8 |
| 2/6/24 |
Cosets. Lagrange's Theorem. |
note 9, note 10 |
| 2/8/24 |
More cosets. RSA cryptosystem. |
note 11 |
| 2/13/24 |
Normal subgroups. Homomorphisms. |
note 12, note 13 |
| 2/15/24 |
Homomorphisms, continued. Complex numbers and groups. |
note 14 |
| 2/20/24 |
Practice session |
problems |
| 2/22/24 |
Exam 1 |
|
| 2/27/24 |
More isomorphisms. Kernels. |
note 15 |
| 2/29/24 |
1st Isomorphism Theorem. Symmetries of a cube. |
note 16 |
| 3/5/24 |
More isomorphism theorems. Cayley's Theorem. |
note 17, note 18 |
| 3/7/24 |
Classification of finite groups. |
note 19 |
| 3/12/24 |
Spring break |
|
| 3/14/24 |
Spring break |
|
| 3/19/24 |
Introduction to rings. |
note 21, note 22 |
| 3/21/24 |
Homomorphisms and kernels. |
note 23 |
| 3/26/24 |
Principal ideals. |
note 24, note 25 |
| 3/28/24 |
Practice session |
problems |
| 4/2/24 |
Exam 2 |
|
| 4/4/24 |
Rings and geometry. |
note 26 |
| 4/9/24 |
Prime and maximal ideals. |
note 27 |
| 4/11/24 |
Factorization in the Gaussian integers. |
note 28 |
| 4/16/24 |
Primes as the sum of two squares. |
note 29 |
| 4/18/24 |
Field extensions, algebraic numbers. |
note 30, note 31 |
| 4/23/24 |
Interlude: Rubik's Cube Group. Degrees of field extensions. |
note 20, note 32 |
| 4/25/24 |
Intro to Galois Theory. Solvability for roots of polynomials. |
note 34, note 35 |
|