Professor: George
Seelinger

Office: Stevenson 313C (Enter through STV 313)

Phone: 438-8781

email: gfseeli@ilstu.edu

Office Hours: W 2:00-2:50,
R 11:00-11:50, F 12:00-12:50

**TEXT:** *Abstract
Algebra, An Introduction*, by Thomas W. Hungerford, 2nd Edition,
Brooks/Cole, 1997.

**COURSE WEB PAGE:** http://www.math.ilstu.edu/gfseeli/m236121/

Here
you will find links to the current assignments, possible posting of hints, and
other resources available on the WEB.

Some Definitions and Theorems (To be updated throughout the semester.)

**ABOUT THE COURSE:** In this course we will cover most of Chapters 1-6
and Chapter 12 of the text with occassional excursions in the
appendices. We will start by examining the familiar set of integers. Our
emphasis in looking at the integers is twofold. First, we want to emphasize the
algebraic properties of the integers. Second, we want to try to develop the
students' ability to understand theoretical mathematical arguments and to be
able to write coherent mathematical arguments. Almost all of the mathematical
arguments you will be expected to understand and to write take the form of
mathematical proofs. Once we gain some experience in the relatively concrete
setting of the integers, we will develop some understanding of abstract rings
and the functions between them. In this less familiar context, the skills of
mathematical argumentation that we developed earlier will become more important
to the understanding of these topics. To further our understanding of rings, we
then look at the arithmetic of polynomials as an example of a ring that is not
the integers. We will then begin the study of quotient fields and the
construction of field extensions, the depth of this coverage will depend on the
time available at the end of the course.

**GRADING:** In this course you will be graded on your performance on
three one hour in class exams, a comprehensive final exam, and weekly homework
assignments. The relative weights of these components will be

Test 1 100 pts (Friday, Feb. 17) Test 2 100 pts (Friday, Mar. 23) Test 3 100 pts (Friday, Apr. 27) Homework 150 pts Classwork 50 pts Final Exam 200 pts (Tuesday, May 8, 3:10 - 5:50 pm, STV 229)

Exam I Solutions

Exam II Solutions

Exam III Solutions

Final Exam Review Problems

**HOMEWORK:** Homework assignments will consist of four or five proofs a
week. Assignments will be given in class and will be due each Thursday by the
beginning of class. As developing your skills at writing mathematical arguments
is one of the central goals in this course, doing as much as you can on each
homework assignment is essential for a good grade. NOTE: In general you should
not expect to be able to do a good job on a homework assignment if you start the
day before it is due. Some problems may require numerous attempts before you
will be able to solve them. As well as the formal written assignments given in
class, it will be necessary for most students to read and re-read the relevant
sections of the text. For your FIRST READING ASSIGNMENT please read ``Appendix
A, Logic and Proof'' (pp. 493-503) in the text.

**Homework Assignments:**

__Homework #1__, due Jan 26: Sect 1.1, p.6-7 # 5, 6, 8; Sect 1.2, p.13 # 4, 8.

(See proof-writing tips to help with doing your homework.)

__Homework #2__, due Feb 2: Sect 1.2, p. 13-14 # 15, 28, 33; Sect 1.3, p.18-19 # 8, 16.

(HINT: Sect 1.2, # 28: Write *d* in the form *au*+*bv* and consider *cd*.)

__Homework #3__, due Feb 9: Sect 1.4, p. 23 # 10, 13; App D, p. 531 # 16; Sect 2.1, p. 29-30 # 15, 23

__Homework #4__, due Feb 16: Sect 2.2 # 7, 8; Sect 2.3 # 4, 7, 8

Homework #5, due Feb 23. (See the pdf file for this assignment by clicking here.)

__Homework #6__, due Mar 1: Sect 3.1 # 5, 9, 18, 22 (Note that for Problem 22, it might be useful to review the properties of logarithms. A review sheet can be found
here.)

__Homework #7__, due Mar 8: Sect 3.2 # 5, 11, 12, 14, 22

__Homework #8__, due Mar 22: Appendix B # 27, 35; Sect 3.3 # 10, 19, 25

__Homework #9__, due Mar 29: Sect 3.3 # 28, 31; Sect 4.1 # 5, 12

__Homework #10__, due April 6: Sect 4.2 #6, 14, 15

__Homework #11__, due April 12: Sect 4.3 #9, 12, 14, 21

__Homework #12__, due April 19: Sect 4.4 #8, 19, 24; Sect 4.5
#5, 6

__Homework #13__, due April 26: Sect 4.5 #18; Sect 5.1 #10, 12; Sect 5.2 #2, 7

__Homework #14__, due May 4: Sect 5.3 # 1, 5, 8, 9 |
Solutions

**Worksheets:**

Worksheet 1: Euclidean Algorithm (Due Friday, Jan 27).

Worksheet 2: Arithmetic of Remainders (Due Thursday, Feb 9).

Extra Credit Worksheet: Ring Isomorphsisms.

Worksheet 3: Irreducibility in **Q**[*x*] (Due Friday, April 20).

Worksheet 4: Field Extensions and Factoring (Due Thursday, May 3) | Worksheet Solutions

**Other Files/Handouts:**

Some Addition and Multiplication Tables for **Z**_{n}

Solving Some Congruences in
**Z**_{n}

RSA Encryption Example using Sage

Statement and Proof of Lemma 3.3

The Quaternion Example from Feb 29^{th}

Example of Euclidean Algorithm in **Z**_{5}

**Algebra on the WEB:**

Sage Math Online

The Development of Ring Theory
(Article by: J J O'Connor and E F Robertson)

First 1000 Primes

The Great Internet Mersenne Prime Search Project (GIMPS)

The Millennium Problems posed by the
Clay Mathematics Institute.