Math 200B: Algebra II (winter 2014)

Course description

This is the second quarter of the three-part graduate algebra sequence. In this quarter, we will cover primarily module and field theory as in Chapters 10-14 of Dummit and Foote's text, Abstract Algebra.

Instructor:	Alina Bucur (alina at math dot ucsd dot edu), APM 7131
Lectures:	usual lecture time: MWF 10-11am, APM 5402; makeup lecture time: Tu 11am-12pm APM 6402 (~~7218~~) starting Feb 4
Office hours:	Tu 3:30-4:30pm, W 4-5pm and by appointment

TA:	Susan Elle (selle at math dot ucsd dot edu), APM 6422
Office hours:	Th 12-2pm and by appointment

Books:	D. S. Dummit and R. M. Foote, Abstract algebra (the main textbook) D. Hungerford, Algebra M. Artin, Algebra J.S. Milne's notes on Galois theory on this website
Prerequisites:	MATH 200A.
Homework:	Homework will be assigned weekly (most weeks) and due on Fridays. The assigned problems will be posted on the class website. Only selected problems will be graded, but you are responsible for completing and understanding all problems. You are free to discuss the homework problems in general with the professor, the TA, or each other, but your final write-up of the problems must be your work alone. If you actually submit solutions that are not your own work, for example directly copying from an online solution bank, you will not get credit and I will have to report you for academic dishonesty.
Exams:	There will be one in-class midterm, tentatively scheduled for Monday, February 24. The final exam will be Friday, March 21, from 8:00-11:00am.
Grading:	Your grade will be roughly based on a final score given by using the following percentages: Homework 25%, Midterm 25%, Final Exam 50%. However, letter grades for graduate students are really just advisory. Your grade in this class is meant to reflect how your current performance corresponds to your likely result on the qualifying exam to be held next year: A = PhD Pass, A- = Provisional PhD Pass, B+/B = Master's Pass, C or less = not likely to pass the qual.

Announcements

The first meeting of the course will be Wednesday, Jan 8. (No class on Monday 1/6, but we might have to make up this lecture.)
I need to fix a time slot for make-up lectures. Please fill out the Doodle poll with your availability. Note that the poll references a specific week, but I am really asking about your general weekly schedule.
The midterm is scheduled for ~~Friday Feb 28~~ in class.
There will be no lectures the week of Jan 27-31 and on Feb 26, 28. Instead we will have makeup lectures on Tuesdays 11am-12pm. The dates for the makeup lectures are Feb 4, 11, 18; March 4, 11.
Makeup lectures will be in APM 6402 starting on Feb 11.
Per popular demand, the midterm will be Monday, Feb 24 in class.
HW7 due date has been extended to Monday 3/3.
I will have office hours on Thursday 3/20 (the day befroe the exam!), 2-5pm.
The final exam is scheduled for Friday, March 21, 8-11am, in APM 5402.

Lectures

W 1/8 Modules: definition, examples, homomorphisms, submodules, quotients, isomorphism theorems.
F 1/10 Direct sums and products, free modules (definition, examples, properties).
M 1/13 Internal direct sums, the R/I-module structure of M/IM, universal property of the free module on a set.
W 1/15 Short exact sequences, split short exact sequence and free modules.
F 1/17 Categories, universal objects, products.
W 1/22 Coproducts, functors, exactness. Hom_R(X,-) is left exact.
F 1/24 Projective modules (definition and characterization), examples, free modules are free, every module is the quotient of a projective.
M 2/3 Injective modules, Baer criterion, divisible groups.
T 2/4 The category of R-modules has enough injectives. Tensor product (definition, examples).
W 2/5 Tensor product and bimodules, extension of scalars, tensor product of homomorphisms.
F 2/7 Associativity, adjointness property of the tensor product, M⊗- is right exact.
M 2/10 Flat modules, projectives are flat, examples. Started modules over PIDs (torsion and torsion-free modules)
Tu 2/11 Modules over PIDs: submodule of a free module is free, torsion-free finitely generated modules are free. Presentation matrix.
W 2/12 Modules over PIDs: started the proof of the structure theorem.
F 2/14 Modules over PIDs: finished the proof of the structure theorem. Modules over F[X].
Tu 2/18 Rational canonical form, Jordan canonical form.
W 2/19 Finite and algebraic field extensions.
F 2/21 Algebraically closed fields.
M 3/3 Algebraic closure of a field.
Tu 3/4 Splitting fields, normal extensions.
W 3/5 Separable polynomials, separable extensions.
F 3/7 Perfect fields, primitive element theorem.
M 3/10 Galois extensions: definitions, examples. Artin's theorem for a finite group of automorphisms.
Tu 3/11 The main theorem of Galois theory, abelian extensions.
W 3/12 Applications of Galois theory. Quadratic, cubic and quartic extentions.
F 3/14 Solvability. Fundamental theorem of algebra.

Homework

Due by 5pm on the posted date in the relevant hw box in the basement of APM.

HW 1 due Friday, 1/17
from Dummit and Foote

10.1: 8, 9, 21
10.2: 6, 9
10.3: 2, 16, 17

I also suggest you look over exercises 24 and 27 in Section 10.3, as they give valuable insight into free modules, but you don't need to turn those in.

HW 2 due Friday, 1/24 pdf

HW 3 due Friday, 1/31 pdf

HW 4 due Friday, 2/7 pdf

HW 5 due Friday, 2/14
from Dummit and Foote

10.4: 11, 14, 15, 16, 17, 27
10.5: 9, 20, 21
Reading assigmnent: 12.1-12.3, reviewing linear algebra from Chapter 11 as needed. In contrast to the text, we will not emphasize algorithmic aspects of computing the rational or Jordan canonical form.

HW 6 due Friday, 2/21 pdf

HW 7 due Monday, 3/3 pdf

HW 8 due Monday, 3/10 pdf

HW 9 due Friday, 3/14 pdf

Exams: midterm, final