# 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.

### 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. HomR(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