Math 454 - Syllabus

 THE UNIVERSITY OF HAWAI'I AT HILO - MATHEMATICS DEPARTMENT

Course Description:

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.

Learning Outcomes:

The successful student will be able to:
- Prove various theorems about divisibility and congruence
- Prove various theorems about prime numbers and factorizations
- Show that a set with two operations is or is not a ring
- Perform arithmetic in rings of polynomials, including factorization
- Demonstate when a polynomial is reducible or irreducible over a ring
- Understand congruence classes in a ring of polynomials
- Understand Ideals and Quotients of Rings by ideals

Homework:

This 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
in 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:

The 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 appropriate software.

The final grade will be based largely on homework, midterms, and final.
Furthermore, class participation (e.g., asking questions in class, going to the Math Lab, going to
office hours, asking Bob email questions, etc) will also be used in this computation.