We are using an excellent 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 theory of rings before introducing 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 one half of this chapter. Next semester we will spend several weeks on group theory, then turn to the study of specific families of rings.
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.
|Ch. 1,2||The integers: divisibility, primality. Congruence classes and arithmetic, the rings Z_m||3 weeks|
|Ch, 3||General rings, and homorphisms||2 weeks|
|Ch. 4||Polynomial rings, divisibility and irreducibility.||2 weeks|
|Ch. 5||Congruence classes and arithmetic in polynomial rings.||2 weeks|
|Ch. 6||Congruences classes and arithmetic for general rings.||2 weeks|
Written assignments should be carefully and neatly presented.The relative weights of the work are given below.