Syllabus 150B: Modern Algebra
Winter 2005

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Lectures: MWF 2:10-3:00am, WELLMN 1   crn: 80624
Discussion section: T 2:10-3:00pm, WELLMN ?
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: most weeks: Monday : 4:30-5:30pm, Wednesday : 3-4pm
in 579 kerr.

T.A.: Brian Wissman wissman@math.ucdavis.edu
Office hours: Monday 10:30-11:30am; Tues 11am-noon; in 475 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. In class.
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 -- February 9, Wednesday.

Final exam: Saturday, March 19 at 8:00am. 1? Wellman
There will be no make-up exams!
Grading: The final grade will be based on: Problem sets (and occasional quizzes) 25%, Midterm 30%, Final 45%
Prerequisites: A proof-writing class (such as 108), and a good linear algebra class (such as 22A or 167) , and 150A.
Web: http://www.math.ucdavis.edu/~vazirani/W05/150B.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   Solutions to final from 150A  
New handouts: on the dihedral group  
a long (8page) handout on rigid motions including lots of notation, and ways of expressing and composing them. this one was NOT handed out in class so you must print it for yourself.  
and a shorter summary of rigid motions and drawing them.  
This handout on finite subgps of O_2 might also be useful.
a copy of page 173 with all the wallpaper patterns, with the point groups noted, and also some red dots showing pts you can rotate around 2 pi/ n radians, as well as some red lines of reflection (or glide reflection).
a more detailed explanation of the point groups of the patterns on page 173. see in particular number 9 for an idea of how to go about finding the point group methodically.

Homework


Homework 1: , due January 12
Solutions by Brian: pdf

Homework 2: , due January 19
Solutions by Brian: pdf
Warning: Artin multiplies permutations (cycles) right to left instead of left to right. i prefer we multiply them as one would functions. so fg(i) = f( g(i)) = f ° g (i) .

Homework 3: , due Jan 26
Solutions by Brian: pdf

Homework 4: , due Feb 2
Solutions by Brian: pdf


Study groups If you are interested in forming a study/hw group w/ your peers, their contact info is here.

Homework 5: , due Feb 9
In doing the HW, if you find the Q^T A Q stuff confusing, you might want to look at old handouts: handout on basis notation like [T]_B^B , a second handout on change of basis ,

Solutions by Brian: pdf

Midterm!!! is Feb 9, in class
Solutions by MV: pdf


Homework 6: , due Feb 16

Solutions by Brian: pdf

Homework 7: , due Feb 23
Solutions by Brian: pdf

Homework 8: , due Mar 2
Solutions by Brian: pdf

While reading about rings and in particular Z/nZ, you might revisit this handout . (i had trouble uploading, so this link might work better.

Homework 9: , due Mar 9

Solutions by Brian: pdf

Homework 10: , try to think about them by Mar 14 ; they are just to reinforce lecture


Final Exam: Sat, Mar 19 - 8:00-10:00 am, 1 Wellman

As you study for the exam, here are some extra problems. In addition to what's listed in the header, also section 5 of Ch7, but not Ch8, and all of Ch 10 that was covered in class/HW. (and you can skip question 7)

Content of the lectures:

The class is based on Chapters 5-10 of Artin's book. Topics to be discussed include:

5. Symmetry
Symmetry of plane figures
The group of motions of the plane
Finite groups of motions
Discrete groups of motions
Group operations (review)
Operation on cosets (review)
The Counting Formula (review)
Finite Subgroups of the rotation group (Tetrahedral group)

9. Group Representations (if time permits)
Definition of a group representation
G-invariant subspaces and irreducible representations
Characters
Permutation representations and the regular representation
One-dimensional representations
Schur's Lemma, orthogonality relations

7. Bilinear forms
Definition
Symmetric forms: orthogonality
the Geometry associated to a symmetric form
Hermitian Forms
the Spectral Theorem

8. Linear Groups
The classical linear groups
the special unitary group
orthogonal representation of SU2

10. Rings
Definition
formal construction of integers and polynomials
Homomorphisms and Ideals
Quotient rings and relations in a ring
Integral domains and fraction fields (if time permits)
maximum ideals (if time permits)