Professor: George
Seelinger

Office: Stevenson 313C (Enter through STV 313)

Phone: 438-8781

email: gfseeli@ilstu.edu

Office Hours: WF 1:00-1:50
R 11:00-11:50

__Finals Week Office Hours:__

Monday, Dec 8: 2:00-3:00

Tuesday, Dec 9: 2:00-3:00

Wednesday, Dec 10: 11:00-12:00

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

**COURSE WEB PAGE:** http://www.math.ilstu.edu/gfseeli/m236084/

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

**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, Sept. 26) Test 2 100 pts (Friday, Oct. 24) Test 3 100 pts (Thursday, Nov. 20) Homework 150 pts Classwork 50 pts Final Exam 200 pts (Wednesday, Dec. 10, 1:00-3:40 PM)

Exam 1 | Solutions to Exam 1 (in PDF
format)

Exam 2 | Solutions to Exam 2 (in PDF
format)

Exam 3 | Solutions to Exam 3 (in PDF
format)

Final Exam Review Sheet (in PDF format)

**HOMEWORK:** Homework assignments will consist of about 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 Aug 28: Sect 1.1, p.6-7 # 5, 6, 8; Sect 1.2, p.13 # 4, 8.

Homework #2, due Sept 4: Sect 1.2 #1 dfh (Use the Euclidean Algorithm to find *d* =
gcd(*a*,*b*)
and integers *u*, *v* such that *au* + *bv* = *d*.), 16, 28;
Sect 1.3 # 16, 23

Solution to Sect 1.3 #23 (in PDF format)

Homework #3, due Sept 11: Sect 1.4, #7; p.531 #11; Sect 2.1, #11, 16, 23.

Homework #4, due Sept 18: Sect 2.2, #3, 8; Sect 2.3, #4, 7def, 9.

Homework #5 (in PDF format), due Sept 25.

NOTE: *You can use MAPLE in the Williams 1G lab to help with this assignment. You could also go to Sage Notebook (beta version) and create an account to use this online system free of charge. Sage differs from MAPLE in that you don't use colons or semicolons, and you use the notation "%n" to mean "mod n". You also have to click on "evaluate" to get it to compute the given value. (Don't worry if you get security warnings about certificates. You should be able to trust this site.)*

Homework #6, due Oct 3: Sect 3.1 # 5, 10, 11, 20, 22

Homework #7, due Oct 9: Sect 3.2 # 5, 9, 11, 23 and the following:

Let *R* and *S* be integral domains. What are the units of
*R* x *S*? What are the zero divisors of
*R* x *S*?

Homework #8, due Oct 16: Appendix B, # 25, 26, 27; Sect 3.3 #17 | Solutions (in PDF format)

Homework #9, due Oct 23: Section 3.3 # 20, 25, 27; Prove that the ring
*L* of Problem 22 of Section 3.1 is isomorphic to **R**, the
ring of real numbers.

**Worksheet on Section 3.3 for Friday, Oct. 17** | Solutions to worksheet

Homework #10, due Nov 6: Section 4.1 # 5, 12, 17; Section 4.2 # 5, 14 | Solutions

Homework #11, due Nov 13: Section 4.3 # 12, 22; Section 4.4 # 8, 16, 19

Homework #12, due Nov 21: Section 4.4 # 7, 12, 18; Section 5.1 # 7, 12 (HINT: On Problem
18(a) try technique similar to the one in the example on p.110. On Problem 18(c), try
reducing modulo 3 and use Theorem 4.24.)

Homework #13, due Dec 4: Section 5.2 #7, 14; Section 5.3 # 5, 6, 9 | Solutions (in PDF format)

**Algebra on the WEB:**

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.