 Overview
Abstract algebra arose from the problem of trying to find the roots of
an equation using radicals. For example, we all know how to find the
roots of any quadratic equation by using the quadratic formula. There
are generalizations of this to third degree and fourth degree
polynomials. One of the more remarkable results of abstract algebra is
that there is no such general solution for fifth degree
polynomials! No matter how hard you try, you can never find a general
formula (or formulas) involving addition, subtraction, multiplication,
division and taking radicals that works for all fifth degree polynomials.
From these beginnings, abstract algebra has found many
applications, including error detection schemes, the RSA encryption
algorithm, solving Rubik's cube, characterizing all periodic wallpaper
borders, and applications in chemistry. This semester we will study
the three major constructions in abstract algebra, groups, rings, and
fields, although we will focus on group theory and ring theory.
We're going to cover alot of material this spring, and you are going
to work hard, but it will also be alot of fun.
I'm really looking forward to this semester.
As a general rule, you should expect to spend 8--10 hours per week outside of class
working on Abstract Algebra.
There will be some weeks
where you spend more time (e.g. working on exams and preparing your
presentations),
but there may be some weeks where you spend less.
Evaluation
Your final grade will be determined by
| Homework | 35% |
| Two Inclass Presentations | 20% |
| Two Takehome Exams | 20% |
| Comprehensive Takehome Final Exam | 15% |
| Quizzes | 10% |
Homework
A major emphasis in this course is that you learn how to
write rigorous and precise mathematical proofs. This can be a
challenging endeavor and may require several iterations, but the process will not only aid your
mathematical development but can also great improve your clarity of
thought in other disciplines and areas as well.
With this emphasis, your homework should be precise, comprehensible,
completely justified, and written in complete sentences.
Most of the homework problems will be worth 5 points, and the possible
grades will be 5, 4, or No Grade. A few of the problems may be worth 10 points,
and the possible grades will be 10, 9, 8, or No Grade.
After I have returned the homework, I will allow you one opportunity
to rewrite any problem. However, you must turn in your rewrite, along with your original
paper, within two class meetings of when I return the homework to the class. For example,
if I return the homework on Monday, you cannot turn in your rewrite after Friday.
An important aspect of your mathematical development is that you learn to discuss
mathematics with others and collaborate on problems.
The homework assignments will alternate between Individual
assignments and Group assignments.
On the group homework assignments, you will work in groups of two
and turn in one paper.
It is extremely important that both of you understand every solution
that your group produces. On each assignment, one student will
be designated as the
primary author who writes-up the solutions, and the role of primary
author must alternate between the members of the group.
For the Individual assignments, I encourage you to work with other
students, but each person must turn in a separate paper.
Here are a few guidelines for the presentation of your homework. If you do not
follow these, I will return your homework to you ungraded!
-
Your writing must be clear and legible. \newline
- Do not turn in your first draft. You should revise, polish
and rewrite your solutions. \newline
- If you write in pen, there should be no scratch-outs. \newline
- Do not turn in paper torn from a spiral notebook with
ragged edges.
The homework is due in my office by 4:00, usually on Wednesday.
Be aware that
Late homework is not accepted!! No exceptions!!
Presentations
You will each give two fifteen minutes presentations during the
semester. The first one will be on a general mathematical topic of
your choosing (I can recommend some good places to look for topics),
and the second will be on
a topic from, or application of, abstract algebra that we have not
discussed in class. I'll give you detailed instructions on my
expectations for these presentations.
Exams
You will have two open book, open note takehome exams during the semester.
I will give you at least one week to work on each exam. See the
syllabus for the due dates.
The final will be a takehome exam and is due Saturday, May 15.
Quizzes
You will have four quizzes during the semester that will consist of
definitions and very basic proofs. I will let you know what your
responsibilities are for each quiz. These are intended to be pretty
low-stress activities.
Hudson River Undergraduate Mathematics Conference
The Sixth Annual Hudson River Undergraduate Mathematics Conference
will be held at Siena College in Loudonville, New York on Saturday,
April 17. The two years that I have gone to this conference, there
have been about 400 undergraduates in attendance, many of whom gave
talks.
I would really like for as many of you as possible to give a talk at
the HRUMC. As an incentive, if you present a talk you will receive an
extra 5% added to your final grade.
Class Attendance
Although class attendance is not a specified percentage of your grade,
I will keep a class roll to help me determine borderline grades at the
end of the semester. If you do miss class, you are responsible
for the material that was covered.
Getting Help
Please come see me during my office hours! If you have a conflict
and cannot make my office hours, please call or email me and we can set up
an appointment for another time.
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