Abstract Algebra II 5211Q - (Spring 2018)

Instructor: Ralf Schiffler
Office: Mont336
Phone: (860) 486-8381
E-mail: schiffler at math dot uconn dot edu
Schedule:
 Tu 2:00-3:15 in Mont 313
 Th 2:00-3:15 in Mont 245
Office Hours:
 Tu 3:15-4:15, Wed 3-4

Midterm Exam: Tuesday March 6, 6-8 pm in Mont 313
No Final Exam.
Projects:
Here is a list of projects from which you can choose if you prefer to make a presentation in class :) instead of a third homework assignment :(
Once you pick a project your name will show up next to it. Of course you can't do one that has already been taken by somebody else. If your favorite one is not available anymore, but you would like to do something in that direction please let me know, so I can come up with another project in the same area. You can also suggest your own projects! It must be sufficiently related to the course and must have sufficient content. [DF ...] means Dummit and Foote.

 
Project
Student
Explain the concept of equivalent categories [DF Appendix II]
Connor
Jacobson radical, Artinian rings [DF 16.1, Prop. 1 & Thm 3]

Jacobson radical of path algebras = ideal generated by arrows [ask me for notes]
Cory
Representations of a group G and modules over the group ring FG, [DF 18.1]
Jonathan
Homological algebra, Cohomology, Ext [DF 17.1]

Affine algebraic sets and coordinate rings [DF 15.1, p. 658 - ...]
Zoe
Projective modules over path algebras [ask me for notes]  
Straightedge and compass constructions [DF 13.3]

Transcendental extensions [DF 14.9]
Tristan
Every module is contained in an injective module [DF 10.5 Thm 38]


In some of the presentations you will explain a part of a theory, give definitions, examples and applications, in others you will prove a theorem in a context that we are already familiar with.

Each presentations will be (at most) 30 minutes long.

You will receive a grade for your presentation, which will depend on the content, clarity and mathematical correctness of your exposition, as well as on your ability to finish on time. The presentations are no oral exams but you should know what you are talking about.




Description:
Module Theory: Tensor product, exact sequences, projective modules, injective modules, tensor algebras, modules over PID;
Field Theory: Algebraic extensions, splitting fields, seperable extensions;
Galois Theory: Fundamental theorem, finite fields, solvable and radical extensions, insolvability of the quintic

Prerequisites: Math5210 Abstract Algebra I

Textbook:  
Dummit and Foote, Abstract Algebra, third edition, Wiley and Sons.
QA162.D85 2004.


This is the book I will use most of the time.
image

Other textbooks:
 
... because sometimes it is nice to see things from a different point of view ...

- Lang, Algebra, QA154.3.L3 2002
- Godement, Algebra, QA155G5913
- Notes on tensor products over commutative rings by Keith Conrad, part 1 and part 2.
- Handout on categories.

Course Grade:
One midterm exam plus either 3 homework assignments or the first 2 homework assignments and one presentation (30 minutes) at the end of the semester.

each homework or presentation
25 %

Midterm

25 %



Practice Exercises: Here is a list of practice exercises from which you can choose. These will not be handed in.

Section
Exercise

Section
Exercise
10.4
1-7, 11-13, 17-19, 21, 23-25, 27



10.5
1-7, 9-11, 20-22, 27, 28



11.5
1, 4-6, 9, 13, 14



12.1
1-3, 5, 6, 8, 20, 21



12.2
5, 7, 14, 18, 19



12.3
1, 2, 21-23