Syllabus

MATH 322 Algebra II Spring 2004

Instructor: Richard Stark, SB 172, ext. 4371, e-mail: rkstark@smcm.edu, rstark@gmpexpress.net

Time and Place: Tuesday, Thursday 10:00 - 11:50

Text: I. N. Herstein  Abstract Algebra

Contents

In the first semester of the course we learned about groups, algebraic structures in which one operation was defined, and about their structure preserving mappings which we called homomorphisms. This term we shall study algebraic structures with two operations, an addition and a multiplication. Such structures are called rings, integral domains, and fields. We investigate their properties and discuss mappings which preserve both of the operations. These are again called homomorphisms. The kernel of a homomorphism is called an ideal and is a special kind of sub ring. Ideals give rise to quotient rings which in turn induce “natural” homomorphisms. Maximal ideals have quotient rings that are fields (if the underlying ring is commutative and has a one). In polynomial rings we define divisibility and develop a theory of divisibility that is similar to that encountered in number theory – the analog to a prime number is called an irreducible polynomial. Of special interest are polynomials with rational coefficients. Just as the integers can be extended to the field of rational numbers, any integral domain has a quotient field. After an excursion into vector spaces, fields are discussed in more detail. Extension fields and, in particular, finite extensions are explored and a theory of constructibility is developed. Finally, extension fields in which a given polynomial has a root will be formed.

Classes

The structure of classes will be discussed in the first class meeting. My hope is that most of the material will be presented by the students in the class.

Jan Hilmar will be teaching assistant for this class. He will attend the classes, help you in preparing for classes and working problems, and will conduct tutoring sessions when needed.

If you need additional help, come and see him or me.

Tests

Testing and evaluation will be discussed in class.

For the final exam date and time – see the college examination schedule