Syllabus 150B: Modern Algebra
Winter 2007

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Grades will be posted on MyUCDavis. Check regularly for errors. You have 2 weeks after the posting (so essentially 3 weeks after a HW is due) to catch errors and bring them to our attention.
Lectures: MWF 10:00-10:50am, OLSON 106;        crn: 50846
Discussion section: R 10:00-10:50am,  OLSON 106
Instructor: Monica Vazirani, MSB 3224, phone: 752-2218, mjvazirani@ucdavis.edu OR (but only if it is really really urgent, otherwise please use the first address) vazirani@math.ucdavis.edu
In ALL emails, please put "Subject: 150B" or I will delete your email as suspected spam.
Office hours: Wednesdays 3:10-4pm. These are my "primary" office hours.
I will have "secondary" office hours on Mondays, which will often be 3:10-4pm, but occasionally 11:45-12:30. (Secondary since I'm also teaching 16B so they "get dibs" that day. ) (On Jan 29, Feb 12 they will have to end at 4 SHARP. Other days they can run over a bit.);

T.A.: Alexander Papazoglou, MSB 3110 papazoga@math.ucdavis.edu
Office hours: Tue 3:30-4:30pm, Thurs: 11:00-12:00am. He'll take a poll and maybe change them later.
Text: Michael Artin, Algebra, published by Prentice Hall, 1991. This is on reserve in Shield library.
Problem Sets: There will be weekly homework assignments, generally handed out on Friday, due the following Friday (or Monday??? comments?) IN CLASS. Or at worst under my office door by 11am.
You are encouraged to discuss the homework problems with other students. However, the homeworks that you hand in should reflect your own understanding of the material. You are NOT allowed to copy solutions from other students or other sources. If you do collaborate with students on a problem, please write their names at the end of your homework (collaborators: Alice, Bob, Carol, etc.). No late homeworks will be accepted. Solutions to the problems will be discussed in the discussion section. This is also a good forum to get help with problems and to ask questions!
Exams: Midterm -- Feb 7, Wednesday.

Final exam: Friday, March 16 at 8:00 am in ?? 106 Olson
There will be no make-up exams!
Grading: The final grade will be based on: Problem sets (and occasional quizzes) 25%, Midterm 25%, Final 50% (or 25-30-45 if that is more favorable)
Prerequisites: 150A, A proof-writing class (such as 108), and a good linear algebra class (such as 22A, 67, or 167).
Web: http://www.math.ucdavis.edu/~vazirani/W07/150B.html

Online resources

Peter Scott's group theory tutorial. (revised link) Each subsection has a quiz at the bottom of the page.

Introduction to permutations, a pdf file, especially the first 10 pages. (this link seems to be inactive!)
There's a brief wikipedia entry on permutations, till I relocate this file.

Introduction to group theory

Here are two links to a book whose chapter on rings is pretty good. A fellow student pointed them out. http://www.math.miami.edu/~ec/book/
http://www.math.gatech.edu/~cain/textbooks/onlinebooks.html

Also, for some of the SU_2 stuff, Wikipedia has this interesting entry Hopf fibration

Other textbooks.

I REALLY like Artin's book, as do most professors. Many students find it a bit terse or lacking in computation and like wordier books or books with more pictures. So I'm listing a few to browse through.
1. I like Contemporary Abstract Algebra by Joseph A. Gallian . See also a related website.
2. This book has MANY examples and some applications. It has come highly recommended to me. Algebra: Pure and Applied by Aigli Papantonopoulou
3. Students tend to like (but instructors not so much!): A First Course in Abstract Algebra, by John B. Fraleigh
NEW Worksheet
OLD
Handout on conjugation and cycle notation (Oct 23).
Random other .pdf files available: hw0--symmetries of square and hexagon , handout on basis notation like [T]_B^B , a second handout on change of basis , handout on Z/nZ , extra practice problems , (note, R[x]_3 is another notation for P_3(R), polynomials of degree less than or equal to 3) more practice problems ,

Problem sets ,


Midterm!!! is Feb 7

Final Exam: Friday, March 16 at 8:00 am 106 Olson

Content of the lectures:

The class is based primarily on Chapters 7,8,10,11 of Artin's book.  (Gasp!  I'm so sad Ch 9 won't be covered till 150C!) Topics to be discussed include:
 

Week 1
Definition of Bilinear Forms (note: Math 67 covers inner products)
Symmetric forms: orthogonality
the Geometry associated to a symmetric form
Week 2
Hermitian Forms (note: Math 67 covers Hermitian matrices)
the Spectral Theorem
Week 3
The classical linear groups
the special unitary group (note: Math 67 covers unitary matrices)
orthogonal representation of SU2
Week 4
 SL(2)
Abstract fields ; matrix groups and linear algebra over abstract fields (from Ch 3 of Artin)
Definition of Rings
Week 5
formal construction of integers and polynomials
Homomorphisms and Ideals
Quotient rings and relations in a ring
Week 6
Integral domains and fraction fields
maximal ideals
Factorization of integers and polynomials
Week 7
Unique factorization domains, principal ideal domains and Euclidean domains
Gaussian integers
Primes
Week 8
Ideal Factorization
Definition of modules
Matrices, free modules and bases
Week 9
Diagonalization of integer matrices
Generators and relations for modules
Structure theorem for Abelian groups
Week 10
Application to linear operators

(if there is extra time, continue with Artin Ch 11 (modules), or go back and fill in topics such as below)

8.6 The Lie Algebra
8. 8 Simple Groups
10.5 Adjunction of Elements