Math 200b Winter 2015
We will have extra lectures Wednesdays 4-5 in 7218 AP&M, on 2/18, 2/25, and 3/4. The last day of the quarter,
Friday 3/13, class will not be held.
|| 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): (Section 10.1-10.2) 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): (Section 10.2-10.3) 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): (Section 10.3) Direct sums and direct products of modules. The submodule generated by a subset. Free modules and their universal property.
- Lecture 4 (1/12/15): (Section 12.1) 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): (Section 12.1-12.2) 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): (Section 12.2) 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.
- Lecture 7 (1/26/15): (Section 12.3) Jordan canonical form over an algebraically closed field. Relation to generalized
- Lecture 8 (1/28/15): (Section 10.4) Tensor products. The universal property of a tensor product. Existence of the tensor product.
Module structure of M \otimes_R N in case M or N is a bimodule.
- Lecture 9 (1/30/15): (Section 10.4) More on tensor products. Given a homomorphism of rings R to S, we can extend
any left R-module M to get a left S-module S \otimes_R M. If R is commutative and M and N are R-modules, M \otimes_R N
is universal for R-bilinear maps. Direct sums pull out of either coordinate of a tensor product.
- Lecture 10 (2/2/15): (Section 10.4-10.5) R-algebras. The tensor product of two R-algebras is an R-algebras. Example: K[x] \otimes_K K[y]
is isomorphic to K[x,y]. Introduction to homological algebra. Short exact sequences. Split short exact sequences.
- Lecture 11 (2/4/15): (Section 10.5) Right exactness of tensor product. Left exactness of Hom. Projective modules.
- Lecture 12 (2/6/15): (Section 10.5) Projective modules are the same as direct summands of free modules. Examples of projective modules.
Injective modules. E is injective if and only if Hom(-, E) is exact, and P is projective if and only if Hom(P, -) is exact. Flat modules. Projective
modules are flat.
- Lecture 13 (2/9/15): (Section 10.5) More on injective modules. The Baer criterion. Injective modules are divisible. Divisible
modules over a PID are injective. Examples of injective modules. The adjointness of Hom and tensor.
- Lecture 14 (2/11/15): (Section 13.1) Introduction to field theory. Two ways to produce new fields from
existing ones: take an irreducible polynomial f in F[x] and look at F[x]/(f), which is a new field of dimension deg f as a vector
space over F; or take the field of fractions of F[x] to get F(x). Given a field extension F contained in K, the field
generated over F by a single element alpha in K is of one of the two types.
- (2/13/15): Midterm
- (2/16/15): (No class, President's Day)
- Lecture 15 (2/18/15): (Section 13.2) Algebraic extensions. Given F in K in L, we have [L:F] = [L:K][K:F].
The set of algebraic elements in an extension F to K is a subfield of K. If K/F and L/K are algebraic extensions then
L/F is algebraic.
- Lecture 16 (2/18/15): (Section 13.4) Algebraically closed fields. Every field F has an algebraic closure (proof using
- Lecture 17 (2/20/15): (Section 13.4) Given a field F and an irreducible polynomial f, there exists an extension K of F in
which f has a root. Given two roots alpha_1, alpha_2 of f there is an isomorphism F(alpha_1) to F(alpha_2) fixing F. Splitting fields.
Lifting lemma: Given any isomorphism of fields F to F' taking a polynomial f to f', there is an isomorphism of a splitting field E of f over F
to a splitting field E' of f' over F'.
- Lecture 18 (2/23/15): (Section 14.1) Automorphism groups of fields. Examples. Galois extensions. The splitting
field of a separable polynomial is Galois.
- Lecture 19 (2/25/15): (Section 14.2) Characterizations of Galois extensions. If L is the fixed field
of a finite subgroup of aut(K), then K/L is normal and separable.
- Lecture 20 (2/25/15): (Section 13.5) Separable and inseparable extensions. An irreducible polynomial
is inseparable if and only if df/dx = 0; this happens only in characteristic p. Perfect fields. All irreducible polynomials over perfect fields
are separable. Purely inseparable extensions.
- Lecture 21 (2/275): (Section 14.2, 14.4) A finite degree extension has a primitive element if and only if there are finitely
many intermediate subfields. Any Galois extension has finitely many intermediate subfields. Any finite degree separable extension
has a Galois closure and so has a primitive element.