Our last day of class: Fri. Dec. 10
| SCHEDULE | |
| ASSIGNMENTS |
You are familiar with the term "algebra" from several years of math classes, dating back to elementary school. In this course, we will study many things from algebra that are familiar--factoring, fractions, functions--but we will explore deeper questions than you have seen in the past. In particular, we will *define* terms precisely, we will state *theorems* and we will *prove* them. This concern with *truth* makes this a course in pure mathematics. Since we start from simple assumptions and build theory from that basis, this mathematics is also called abstract.
We are using a well written text book in this course. It takes an unusual approach to the subject, starting with structures that you know well--the integers and polynomials--then developing the more general and abstract theory of rings and, finally, groups. The author organizes the material to emphasize central themes in abstract algebra: divisibility, the notion of congruence, the creation of new algebraic structures from known ones and homomorphims (functions that preserve structure). Chapters 1 and 2 discuss these notions for the integers; Chapters 4 and 5 discuss them for polynomial rings. Chapters 3, and 6 develop the same ideas for general rings. Chapter 7 is a new beginning, introducing the more general and abstract theory of groups. We will only cover the first few sections of that chapter.
This year I am introducing something new: the use of a computer software package called sage. This will allow us to compute examples that would be difficult (or even impossible) to do by hand, explore algorithms, and learn a bit of programming.
The main prerequisite is Discrete Mathematics, Math 245. It will be worthwhile to review the fundamentals of logic ("and", "or", "implies", "for all", "there exists", and negation). We will use a variety of proof methods in this course: direct proof, proof by contradiction, and mathematical induction. We use equivalence relations.
| SECTIONS | TOPICS | TIME |
| Ch. 1,2 | The integers: divisibility, primality. Congruence classes and arithmetic, the rings Z_m | 3 weeks |
| Ch, 3 | General rings, and homorphisms | 3 weeks |
| Ch. 4 | Polynomial rings, divisibility and irreducibility. | 3 weeks |
| Ch. 5 | Congruence classes and arithmetic in polynomial rings. | 1 week |
| Ch. 6 | Congruences classes and arithmetic for general rings. | 2 weeks |
| §7.1-5 | Groups. | 3 weeks |
Written assignments should be carefully and neatly presented. You may work with others to solve problems, but the final written work should be done individually.
The relative weights of the work are given below.
| Weekly work | 250 |
| Tests | 450 |
| Final | 300 |
| Total | 1000 |