Field theory
The course meets at 1:30 MWF in BH 235 B. 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 Ernest Schimmerling
on Monday 14 and Friday 18, and by Prof Jeremy Avigad
on Wednesday 16. 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.
Text: Dummit and Foote "Abstract Algebra" (make sure to get the third edition!)
Tentative syllabus (this will firm up as the term progresses):
- Review of material from ``Algebraic structures''.
- Field extensions: algebraic and transcendental elements.
- Degree of an extension, adjoining a root of a polynomial.
- Splitting field of a polynomial.
- Finite fields.
- The Galois group of a field extension.
- The Galois correspondence.
- Applications:
- Proof that there is no general solution for an equation of degree five.
- Theory of symmetric polynomials.
- The transcendence of e and pi.
- There is no ruler and compass construction for squaring the circle.
- The ``Fundamental Theorem of Algebra''.
- (Time allowing) Advanced topics:
- Structure of algebraically closed fields.
- Luroth's theorem.
- Infinite degree Galois extensions.
- Applications to number theory.
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 ring theory and 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 TeX.
- Homework 3 in PDF and TeX.
- Homework 4 in PDF and TeX.
- Homework 5 in PDF and TeX.
- Final in PDF and TeX.
- Homework 1 solutions.
- Homework 2 solutions.
- Homework 3 solutions.
- Homework 4 solutions.