Syllabus

MATH 322 Algebra II Spring 2003

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

Time and Place: TR 10:00 - 11:50, SB160

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 subring. 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, the number of quizzes and homeworks will be discussed in the first class meeting.

Amber Wagner will again be teaching assistant for this class. She will attend the classes, give some lectures

and work problems, and will conduct help sessions when needed.

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

There will be three tests and a final. The tests may be taken in either written or oral form, although at least one must be taken orally and one in written form. If you opt for an oral exam, you will have a second chance in case you are not satisfied with your performance.

Tests

Test 1 -- Thursday, February 13 (Orals by appointment)

Test 2 -- Thursday, March 13 (Orals by appointment)

Test 3 -- Thursday, April 17 (Orals by appointment)

Final exam – see the college exam schedule