Mathematics Department | Illinois State University

## Math 236, Section 2, Elementary Abstract Algebra

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.