Algebra I, Math 5420, Fall 2015


Table of Contents

Class:

Professor:

Overview:

This is a first course in abstract algebra. It serves as an introduction to a very powerful way of thinking, which allows us to derive quite complex concepts from first principles. If things go well, you will remember this semester as an exhilarating intellectual experience. At the same time, you may find that this course can be quite demanding of your time and energy. It will certainly be harder to catch up in this course than it would be in a lower-level mathematics course.

Class participation will be essential to mastering this material, and attendance is mandatory. You will be expected to present material in class. You should notify me if you are unable to attend even a single meeting of the class. Tests will presume familiarity with material presented in class as well as the material in the text.

Prerequisites:

MAT 2030 and MAT 2250 are the formal prerequisites for this course, but you will find that this course operates at a much higher level of mathematical sophistication. MAT 5400 (Number theory) and/or MAT 5000 (Proof Writing) are not prerequisites, but may serve as a gentler introduction to this level of mathematical thought. Similar courses, such as MAT 5410, 5520 or 5600, may also be helpful. None of the material in these 5000 level courses will be required to understand MAT 5420.

Writing Intensive Course:

Undergraduate mathematics majors taking this course will normally be co-enrolled in MAT 5993, Writing Intensive Course in Mathematics, which satisfies a WSU General Education Requirement. A distinctive feature of mathematical writing is the use of formal proofs. As with many other skills, the key to mastery is regular and consistent practice to build confidence and good technique. The section To the Student (pp. xii - xx of the text) contains extremely good advice on these matters.

Advice:

READ THE BOOK . You should read each section before we talk about it in class, then again after class, before doing the homework for the section. In fact, you should read it several times. First, quickly, for an overall idea what the section is about, then in detail, working through the proofs and examples, line by line, to make sure you know why each statement is true. Only after this should you start the homework. You will be pleasantly surprised how much easier the homework is with this sort of preparation. You will certainly understand the material and retain more of it, if you study in this way. Further, you will know the sections well enough to refer to them precisely in your writing. Accurate and precise citations play an important role in mathematical writing.

Definitions play an important role in abstract mathematics. You should memorize them. In everyday life, and in many other subjects, definitions are designed to give you an understanding of what a term means. In mathematics, definitions are designed to carefully isolate the exact properties we need. In fact, to make a definition easier to use, mathematicians streamline them to the bare bones. The meaning of the definition emerges gradually from the theorems which we prove. Similarly, the precise hypotheses of theorems must be remembered in order to use them correctly.

Schedule:

The first class is Wednesday, September 2, 2015, and the last class is Monday, December 14, 2015. The final exam is Friday, December 18, from 8:00 to 10:30 in the usual classroom, 323 State Hall.

We will cover at least the first five chapters of the book, and if time permits, Chapter 6. Here is a schedule of the topics and chapters we will cover, as well as the timing of the tests.

Introduction (Integers
and Functions) 		Chap. 1 and 2, and
Appendix A.1 - A.4 		   Sept. 2 - Sept. 25
Test 1 ( Solutions)    Friday, Sept 25
Group Theory, Part I Chap. 3    Sept. 28 - Oct. 16
Test 2 ( Solutions)    Friday, Oct. 16
Group Theory, Part II      
and Polynomials 		Chap. 3 (cont.) and Chap. 4    Oct. 19 - Nov. 13
Test 3 ( Solutions)    Friday, Nov. 13
Commutative Rings and
Fields 		Chap. 5 and Chap. 6    Nov. 16 - Dec. 14
No class Thanksgiving break    Nov. 25 and 27
Test 4 ( Solutions)    Thursday, Dec. 11
Classes end    Dec. 14
Review    Dec. 16, 9:35 - 10:30, location to be announced.
Final Exam    Friday, Dec. 18
8:00 - 10:30, in 323 State Hall

Requirements and grades:

Each Tuesday, volunteers from the class will be asked to present solutions to selected homework problems. In addition, starting after the first week, class members will be asked to present proofs in class. The items to be proved will be mutually agreed upon, and you will have a week to prepare your presentation.

Class members will be expected to actively participate in these presentations, and in mine. Your participation will determine a quarter of your 'presentations' grade.

Grades will be computed as follows:

4 In-class exams       400
Homework 200
Presentations 200
Final 200
Total 1000

Learning Outcomes:

Upon completion of MAT 5420, you will have made three notable conceptual advances. The first is a shift from understanding based on the particular objects under consideration, to understanding based on their properties. Second, you will move from consideration of individual elements, one at a time, to collections of objects which have a structure of their own. Third, you will come to understand quotient objects, a far more subtle and difficult concept than sub-objects.

As a result of the first shift, you will become fluent with arguments of the form ''if these properties hold, then these others must as well''. The second will build your capacity to move between levels: understanding the behavior of individual mathematical objects by virtue of the properties of the collection(s) they are a part of, and vice versa. You will become able to determine which level is appropriate to the problems you are considering. The third shift will use the first two skills to enable you to understand and work with quotients.

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