UNIVERSITY OF HAWAI'I AT HILO - MATHEMATICS DEPARTMENT
Our course in Abstract Algebra will take various well-known properties
of integers and polynomials and generalize them to a broader, more
abstract class of objects called Rings and Groups. At its heart,
the integers is simply a set where multiplication and addition make
sense and behaves nicely. A ring is simply a generalization of
this. In our course, we will see that various key aspects of the
integers persist when we abstract this set. For example, prime
numbers have a generalization called prime ideals in a general
ring. We will then further explore polynomials and their various
properties and elegance. We will investigate why, when we
restrict ourselves to certain rings, like the real numbers, polynomials
like x^2 + 1 are irreducible; however, when we extend our rings to that
of complex numbers, we can easily factor x^2+1 = (x+i)(x-i). To
better understand this behavior, we will discuss congruence classes in
polynomials rings, when we "forget about" certain parts of polynomials
and only focus on remainders under polynomial division.
The purpose of this course is to obtain a better understanding of the
integers, integers modulo n, rationals, reals, and complex numbers
while identifying the key basic properties that makes arithmetic in
these rings so special.
The successful student will be able to:
- Prove various theorems about divisibility and
- Prove various theorems about prime numbers and
- Show that a set with two operations is or is not a
- Perform arithmetic in rings of polynomials,
- Demonstate when a polynomial is reducible or
irreducible over a ring
- Understand congruence classes in a ring of
- Understand Ideals and Quotients of Rings by ideals Homework:
course will have frequent homework assignments. You are not only
allowed - but encouraged -
to work in small groups for these
assignments. However, each student must turn in homework written
his/her own words and must understand all the content of the HW.
Failure to do so may be
interpreted as cheating.
Homework will, at first, be allowed to be hand-written or typeset, but
eventually, all homework will be required to be typeset in LaTeX.
Homework must be printed out prior to class and turned in at the
beginning of lecture on the due date. LaTeX:
use of AMS LaTeX is not only encouraged but will be
required. LaTeX is the
universally accepted scientific typesetting program that allows
mathematicians to write well-formatted mathematical papers. This
open-source compiler and associated interfaces can be found as
freeware. Please see the below links for how to download the
grade will be based largely on homework, midterms, and final.
Furthermore, class participation (e.g., asking questions
class, going to the Math Lab, going to
office hours, asking Bob email
questions, etc) will also be used in this computation.
grade will be computed using the following weights:
50% - Homework
35% - Midterms & Final 15% -