Syllabus

MAT 3223 – Modern Algebra

Fall 2009

Instructor: Sandy Norman

Office: SB 4.02.32

Phone: 458-7254

Office Hours: 2:00—3:00 p.m. MW and by appointment

e-mail: sandy.norman@utsa.edu

class webpage: http://math.utsa.edu/~fnorman/3233/mat3233.htm

Text: Abstract Algebra: A Geometric Perspective, Ted Shifrin: Prentice Hall, 1995

· ISBN-13: 9780133198317

Prerequisites: MAT 1214 and MAT 3013

I will presume that students taking this course have a certain degree of mathematical “sophistication." That is, you should be familiar with sets and set notation, Venn diagrams, properties of sets, matrices (at least to the level of using them to solve systems of equations, determinants, Cramer’s rule, and such), and the basics of mathematical proof. The course text has a brief discussion of some of these topics and should be reviewed independently before we get deeply into the course.

Course Objectives: There are two primary objectives for the course:

(1) Provide an introduction to the fundamental concepts of abstract algebra that form the foundation of secondary school algebra; and

(2) Build on the basic proof construction skills developed in Foundations of Mathematics in the context of number theoretic, algebraic, and

geometric settings.

Important Dates: …. can also be found at the Academic Calendar site.

First Day of Class ………………. 08/26, Wednesday

Census Date …………………….. 09/11, Friday

Last day to drop via ASAP …….. 10/29 by 8:00am, Thursday

Thanksgiving Holidays………….. 11/26—11/39, Thursday—Sunday (Class will meet on Wednesday 11/25.)

Last Day to Withdraw ………….. 12/02, Wednesday

Study Days ……………………... 12/7-12/8, Monday-Tuesday

Assessments: Graded assessments for the course includes tests, quizzes, class and homework, active participation, and a cumulative final exam.

Tentative Dates*

Test #1 …………………. 20 Wednesday, September 30

Test #2 ……………….… 20 Monday, October 26

Quizzes ………………… 10 occasionally (may be unannounced)

C/H/M-work …………… 15 frequently

Final Exam ……………... 35 Friday, December 11, 1:30—4:00 p.m.

*Except for the Final Exam, specific dates for the assessments may change. These will be updated on this syllabus and announced in class. Check frequently.

Make-ups for tests will be given only with my prior approval or in extremely extenuating circumstances. There will be no make-ups for missed quizzes or classwork.

Students are expected to assist in maintaining a classroom environment that is conducive to learning. To assure all students have the opportunity to gain from time spent in class, students are prohibited from engaging in any form of distraction. Inappropriate behavior in the classroom shall result, minimally, in a request to leave class. Students are expected to be familiar with and abide by principles of scholastic honesty as described by UTSA’s Student Code of Conduct.

Outline of course:

Though I like to present lots of examples, this is basically an abstract course. As such, you will need to gain some facility with abstracting the

underlying properties and structure of algebraic objects (groups and fields, in particular) and recognize and use these properties/structures in a variety of familiar (and perhaps less familiar) mathematical contexts.

Here is a tentative schedule for the material we will cover.

• 8/26: Introduction; integers, induction, binomial theorem [1.1]
• 8/31: Euclidean algorithm [1.2]
• 9/02: Modular arithmetic and congruences [1.3]
• 9/07: Labor Day Holiday, no class
• 9/09: Rings, integral domains, and fields [1.4]
• 9/14: finish up Chapter 1 topics [1] \cbu6

• 9/16: Rationals [2.1]
• 9/21: Rationals à Reals [2.2]
• 9/23: Complex numbers [2.3]
• 9/28: Quadratic/Cubic formulae [2.4]
• 9/30: Test #1

• 10/05: Isometries [2.5]
• 10/07: Euclidean algorithm for polynomials [3.1]
• 10/12: Roots of polynomials [3.2]
• 10/14: Polynomials over Z [3.3]

• 10/19: Ring homomorphisms and ideals [4.1]
• 10/21: Isomorphisms and homomorphism theorems [4.2]
• 10/26: Test #2
• 10/28: Gaussian Integers [4.3]

• 11/02: Vector spaces and dimension [5.1]
• 11/04: Compass and straightedge constructions [5.2]
• 11/09: continued [5.2]
• 11/11: Finite fields [5.3]

• 11/16: Intro to Groups [6.1]
• 11/18: Group homomorphisms and isomorphisms [6.2]
• 11/23: continued [6.2]
• 11/25: Cosets, normal subgroups [6.3]
• 11/30: Quotient group and the symmetric group [6.4]

Miscellany

Being diligent in working, even struggling, with the problems in the book is essential to both the learning experience and your performance on tests. Exactly how much is necessary varies from one individual to another. Many of the problems you encounter will not be computational --- rather, solutions will look more like proofs, and may indeed {\it be} proofs. It takes careful (and logical!) thought, hard work, and mathematical imagination to get really good at solving some of these types of problems. Such things are intinsic and can't really be taught; they arise from experience, hard work, self-motivation, and a certain amount of mathematical and personal maturity.

In class I shall assign a few problems in each section that I believe to be representative or particularly important. These are to be turned in at final examination time, although I reserve the right to check your ongoing work at any time. You may find that in doing some of the assigned problems it will be helpful to do other problems that precede them in a given section.

You may work in groups (I encourage this), but each of you must write up and turn in your own final version of the homework problems individually. If you get stuck on a problem, you should write up what arguments you can muster (if any) and append a brief discussion about the nature of the obstacle and how you might proceed once it is cleared. If you need feedback about the homework beyond what is covered in class, feel free to stop by my office during office hours or at some other agreed upon time.

There are no contingencies for extra credit work. Further, I expect all students to exhibit respect for the instructor and each other. Inappropriate,

distracting, or disruptive behavior is not acceptable in this classroom.

• All homework sets will be collected at the final exam.
• All quizzes and exams will be in class.
• Unless otherwise noted, books and notes will not be permitted during quizzes or exams. Calculators will be allowed at any time.
• The final exam is comprehensive.
• The lowest quiz grade will be dropped.
• No make-up quizzes or exams unless due to exceptional circumstances. Exceptionality is determined by me.

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