Algebra: Doctoral Core II

Math 520B, Fall 1999

Call Number 2764

September 8, 1999

1. Introduction

Math 520A and Math 520B together are intended to provide about 80% of the material needed to prepare a student for the Preliminary Ph.D. Examination in Algebra.

My understanding of the syllabus descriptions of of these courses is given below. In various instances, however, there have been departures from this division of material between the two courses. For that reason an approximate outline of the material to be covered will be set during the first week of the course in consultation with the students in the course who are intending to prepare for the ``prelim''.

2. Linear algebra and this course

The topic linear algebra over a commutative field is not part of the syllabus for the ``prelim'' in algebra since linear algebra is regarded as part of that which is common to all branches of mathematics. But it is important for the student to understand that the theory of modules over a commutative ring is a vast generalization of that subject, so vast, in fact, that almost all of what is learned in that very special case is essentially lost in the general case.

Nonetheless in the context of the ``prelim'' syllabus it is important to understand that the classification of linear endomorphisms of a finite-dimensional vector space, which is equivalent to the classification of similarity classes of (square) matrices of a given size, is entirely subsumed by the structure theorem for finitely-generated modules over a principal ideal domain, a theorem that also subsumes the structure theorem for finitely-generated abelian groups.

3. Syllabus descriptions of Math 520A and Math 520B

Math 520A

The first of two ``core'' courses for the doctoral program. The topics to be covered are finite groups, including the Jordan-Holder theorem and the Sylow theorems, and the theory of extensions of fields, including the various completions of the rational field, algebraic closures, and the Galois theory for finite extensions in arbitrary characteristic.

Math 520B

The second of two ``core'' courses for the doctoral program. This course is an introduction to commutative algebra with a slant toward universal mapping properties and some hint of the notions of modern homological algebra. Topics to be covered include the basic constructions and the various notions of finiteness in the theory of commutative rings and their modules. There should be a careful treatment of the theory of finitely-generated modules over a principal ideal domain and its special cases. On the other hand, it is important for the student to emerge from the course with an appreciation of the fact that not all commutative rings are one-dimensional.

4. Course Requirements

5. Math 520B last year

An outline of the material that I covered during the Fall, 1998 semester is available on the web at
http://math.albany.edu:8000/math/pers/hammond/course/mat520bf98/covered.