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''.
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.
|Final examination||100||Mon. Dec 20 3:30-5:30|
|Midterm test||50||Fri. Oct 22, in class|
|Problem Sets (5 @ 10 each)||50||as announced|
An outline of the material that I covered during the Fall, 1998
semester is available on the web at