Syllabus 150A: Modern Algebra
Fall 2004

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Lectures: MWF 2:10-3:00am, WELLMN 7   crn: 49633
Discussion section: T 2:10-3:00pm, WELLMN 7
Instructor: Monica Vazirani, Kerr Hall 579, 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
Office hours: Monday : 3-4pm, Wednesday : 4:30-5:30pm
this week, i will have an extra office hour on Friday (Dec 10) from 3:10-4pm.
T.A.: Robin Wilson rtwilson@math.ucdavis.edu
Office hours: 466 Kerr; Mon 4-5:30pm and Tues 4:30-6pm.
Robin is having extra office hours (dec 9) Thurs from 4:30-6pm and again on (dec 13) Monday from 2pm until?
Alternate office hours: Section 001 of Math 150A Robin Endelman office hours    week of Dec 6-10th: MW 11am-1pm, TuTh 11am-noon [none on Friday]   in Kerr 673
Alice Stevens   Office hours: Tuesdays and Thursdays 11am - noon   in 473 Kerr  
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 Wednesday, due the following Wednesday. To your TA by noon.
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 -- November 3, Wednesday.

Final exam: Tuesday, December 14 - 4:00-6:00 pm, in ? Wellman
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%
Prerequisites: A proof-writing class (such as 108), and a good linear algebra class (such as 22A or 167).
Web: http://www.math.ucdavis.edu/~vazirani/F04/150A.html

Online resources

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

Introduction to permutations, a pdf file, especially the first 10 pages.

Introduction to group theory

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 , Solutions to quiz 1 (some notation on page 2 for problem 2 is edited here),
midterm exam solutions Solutions to quiz 2

Problem sets

Homework 0: , a handout from class, due Weds, Oct 6.

Homework 1: , due October 13
Solutions by Robin: pdf

Homework 2: , due October 20
It might be helpful to refer to the handout on basis notation like [T]_B^B ,
Solutions by Robin: pdf

Homework 3: , due October 27
Solutions by Robin: pdf (oops, thought i had posted these last week! sorry for the delay. i think robin had posted them at myucdavis meantime.)

Homework 4: , due Nov 3 partial solns so you can study for the midterm are available. i do NOT want to see verbatim solns on your hw4s!
You may wait till 5pm to hand this in in 579 Kerr. (or hand it in in class as usual.)
Solutions by Robin: pdf (revised) (note, on 4.misc.1, the possible lambda are the r-th roots of unity. the description as e^{2 pi i k/ r} is if your field is C . for other fields you just get whatever r-th roots of unity exist in it. (so for R , that would just be 1 or 1 and -1 , depending on if r is odd or even.)

Homework 5: , due Nov 10
Solutions by Robin: pdf

Midterm!!! is Nov 3

Homework 6: , due November 17
Solutions by Robin: pdf Note, page 5 (image and kernel are subgroups) is hard to read. It is rescanned in here. Also there are some minor corrections on the pages, but i think they are clear.

Homework 7: , due November 24
Solutions by Robin: pdf


Homework 8: , due December 1
don't forget to reread the handout on Z/nZ
Solutions by Robin: pdf

Homework 9: , due Dec 8
Solutions by Robin: pdf

Homework 10: , due December 14

Final Exam: Tuesday, December 14 - 4:00-6:00 pm, 7 Wellman

As you study for the exam, here are some extra handouts (courtesy of Dr. Endelman).
extra exercises on groups.
a handout with examples of quotient groups.
Section 01 gave biweekly quizzes. Taking quizzes is a GREAT way to study. These have the soln's on them. If they are going to be of ANY use to you, you need to print out 2 copies, whiting out (or cutting out) the solutions, and practice taking the quiz, giving yourself only 15 minutes. Then grade yourself. quiz6, quiz5, quiz4.
And for other practice quizzes, don't forget Peter Scott's group theory tutorial. Each subsection has a quiz at the bottom of the page.
some extra practice problems .

Final Exam Solutions: pdf.

Everyone have a good break, and I hope to see many of you in 150B.

Content of the lectures:

The class is based on Chapters 1-4, 6 of Artin's book. Topics to be discussed include:

1. Preliminaries
Matrices
Permutations and permutation matrices


3. Vector spaces
Real and complex vector spaces
Abstract fields
Bases and dimensions
Computations with bases
Direct sums

4. Linear Transformations
The dimension formula
The matrix of a linear transformation
Eigenvectors
The characteristic polynomial
Orthogonal matrices and rotation
Diagonalization

2. Group Theory
The definition of a group
Subgroups
Homomorphisms
Isomorphisms
Cosets
Products of groups
Quotient groups
Modular arithmetic

6. Operations of a group on itself
Class equation
Operations on subsets
Sylow Theorems
The Groups of order 12
If there is time, we will cover select parts of Chapters 5 and 9.