Algebra I
The course meets at 10:30 MWF in CFA 212. Office hours are
by appointment only, please send email to
jcumming@andrew.cmu.edu to arrange an appointment.
Homework will be set most Mondays, will be due on the following Monday, and
should be returned graded by the Monday after that.
Late homework will not be accepted under any circumstances, but the lowest
homework score will be dropped. Homework must be submitted by email in
LaTeX format (I want the actual LaTeX, not a PDF or PostScript file generated
from it) by the start of class time on Monday.
I will be away from Pittsburgh for the first week of the term.
Class will be taught by Prof Jeremy Avigad
on Monday 14, and by Prof Rami Grossberg
on Wednesday 16 and Friday 18. Please note that if you have any
questions about the course you should contact me at the email address above.
Prerequisites: A basic knowledge of groups, rings and fields as covered in the CMU
"Algebraic structures" course, and of linear algebra as covered in
the CMU "Linear Algebra" course. See the handouts below for more detail
about what I am assuming that you know. Let me know if you are missing some background, and
I will make a handout and do a lightning recap of the needed material in
class.
Tentative syllabus (this will firm up as the term progresses):
-
Groups (the emphasis is on finite groups for the most part):
- Group actions.
- The Sylow theorems.
- Solvable and nilpotent groups.
- Simple groups.
- Free groups and their quotients.
The treatment of abelian groups is deferred to the ``Rings and modules''
section, where they appear as a special case of modules.
- Rings and modules:
- Modules.
- Structure theory of semisimple rings.
- Structure theory of finitely generated modules over a PID.
- Notherian rings and modules, Hilbert Basissatz.
-
Fields:
- Field extensions.
- The Galois group and Galois correspondence.
- Unsolvability of the quintic.
- Finite fields.
- (Time allowing) Advanced topics: infinite degree Galois extensions,
algebraic closures, structure theory of algebraically closed fields.
There will be a midterm and a final. Grades will be assigned according to a formula
in which (roughly speaking) homework counts 35 percent, the midterm counts 30 percent
and the final counts 35 percent. I encourage collaboration on the homework
but you must write up your solutions by yourself.
- This homework has two goals: getting comfortable with LaTeX and reviewing
some basic concepts in group theory.
- If it is not already installed, install LaTeX on your computer. Here are
some instructions for doing
this on common platforms.
- To test that your installation is working OK, download the
LaTeX source file for Homework 1, and create
a PDF file from it. You should end up with a PDF file which looks
something very like this.
- Read through the sections Absolute Beginners, Basics, Document Structure, Mathematics
and Errors and Warnings in the
wikibooks LaTeX tutorial. You will also find it instructive to compare the source code
in the file "hw1.tex" with the PDF file that was compiled from it.
- Save a copy of the LaTeX file "hw1.tex" which you just downloaded under the name
"hw1_YourFirstName_YourLastName.tex"; this is the naming convention
that will be used throughout the term. Now edit this file to prepare your
homework solutions. Make sure that your LaTeX file compiles (warnings are OK,
errors are not).
When you are finished email this file (as an enclosure, NOT in the body of your email message) to me.
- Homework 2 in PDF and LaTeX formats.
- Homework 3 in PDF and LaTeX formats.
- Homework 4 in PDF and LaTeX formats.
NOTE DUE DATE!
- Homework 5 in PDF and LaTeX formats.
- Final in PDF and LaTeX formats.
- Homework 1 solutions.
- Homework 2 solutions.
- Homework 3 solutions.
- Homework 4 solutions.