Math 200b Winter 2015
|| Lecture Time
|| Lecture Place
|Prof. Daniel Rogalski
|| W 2-3pm or by appt.
|| MWF 1-1:50pm
|| AP&M 5402
| Rob Won
|| AP&M 6321
|| W 3-4, Th 3-4 or by appt.
Course Description and Syllabus:
Math 200b is the second quarter of the three-part graduate algebra sequence. In this quarter, we
will cover module theory and field theory as in Chapters 10-14 of Dummit and Foote's text, Abstract Algebra.
Math 200c in the spring will cover the more advanced theory of commutative rings using the text by Atiyah and Macdonald.
Math 200a is a prerequisite for Math 200b; please come talk to me if you think you are prepared to take Math 200b without
first taking Math 200a.
The detailed syllabus for the course can be found at the following link. It is important that all students
read it since the full course policies may not be gone over in class. Please ask if you have any questions about
Below, we will post short summaries of what was covered in each lecture.
- Lecture 1 (1/5/15): Definition of a (left) module over a ring R. Examples. Modules over a field K are vector spaces
and modules over Z are Abelian groups. Submodules, homomorphisms, factor modules, isomorphism theorems.
- Lecture 2 (1/7/15): The set Hom_R(M,N) of all R-module homomorphisms from M and N is always an Abelian group. If R is commutative it is also a left R-module. End_R(M) = Hom_R(M,M) is always a ring with function composition for the multiplication. End(R) is isomorphic to R if R is commutative. There is a one-to-one correspondence between R-module structures on an abelian group M and ring homomorphisms from R to End_Z(M). If F is a field, an F[x]-module is the same as an F-vector space V together with a linear transformation phi in End_F(V) such that x acts as phi.
- Lecture 3 (1/9/15): Direct sums and direct products of modules. The submodule generated by a subset. Free modules and their universal property.
- Lecture 4 (1/12/15): Statement of the fundamental theorem of finitely genrated modules over PIDs. Outline of proof,
assuming the following technical lemma: If M is a f.g. free module over a PID R, and N is a submodule of M, then there is a
basis m_1, ..., m_n of M and nonzero a_i in R such that a_1m_1, ... a_d m_d is a basis of N for some d less than or equal to n. In particular, N is free.
- (1/14/15): (No class: Professor Rogalski out of town)
- (1/16/15): (No class: Professor Rogalski out of town)
- (1/19/15): (No class: MLK Day)
- Lecture 5 (1/21/15): : Sketch of the uniqueness part of the fundamental theorem of finitely generated modules
over PIDs. Review of how matrices correspond to linear transformations. Beginning of discussion of rational canonical form of a linear transformation.
- Lecture 6 (1/23/15): : More on the rational canonical form. The minimal and characteristic polynomial
and their relation to the rational form and the invariant factors. The Cayley-Hamilon Theorem. Similarity of matrices is
independent of the field.