MATH 440
Abstract Algebra
Syllabus for Fall 2007
Mon & Fri, 10:00 – 10:50am
Wed, 9:00 – 10:50am
326 Gildemeister Hall
Prerequisite: A passing grade in MATH 210.
Text & Calculator: Contemporary Abstract Algebra by Joseph A. Gallian
Course Website: http://course1.winona.edu/eerrthum/math440
Instructor: Dr. Eric Errthum Office: 203L Stark Hall
Winona Email Username: eerrthum Office Phone: 474-5775
Office Hours: See schedule on my home page.
Grading: Quizzes
(10 @ 25 points each, drop two lowest) 200
points
Homework (scaled
as needed) 200
points
Chapter Summary
Paper 100
points
Midterms (3 @ 100
points) 300
points
Final 200
points
--------------
1000
points total
Grades: A = 90% (900 pts), B = 80% (800 pts), C = 70% (700 pts), D = 60% (600 pts)
Exams: There will be three in-class exams and one comprehensive final exam. Exam dates are tentative until officially announced in class. The final exam is tentatively scheduled for Wednesday, December 12, 8:00 – 10:00am.
Quizzes: We will have a short quiz every Wednesday (except for exam weeks). Each quiz will count for 25 points and the lowest two quiz scores will be dropped from your grade.
Homework: Homework will be assigned periodically and will be collected the following Wednesday. You are encouraged to do extra problems and to work on homework together in groups, as long as each member of the group is contributing equally. Each person must hand in their own answers written in their own words.
Late/Missed Work: Late homework or missed quizzes will result in a score of zero. There are no make-up quizzes. Make-up exams will be given at the discretion of the instructor. If you miss class, it is your responsibility to obtain notes and assignments from fellow students. If you have an unavoidable absence, please inform the instructor beforehand.
Motivation: The author of the book has
a lot of good resources on his website: http://www.d.umn.edu/~jgallian/
especially addressing “Reasons
why Abstract Algebra is valuable to math ed majors (and math majors)”.
Paper: Part of your grade will be a paper summarizing a chapter or chapters from the book. Use 12pt font, 1-inch margins, and double-spacing. It will be due at 12:01am, December 1. Please see below for a grading rubric and available topics. Aim to be both concise and thorough. You are not allowed to use other sources beyond the text. Please see your instructor if you are having problems. Note that some topics depend on material that may not be covered in the class until mid-November. Plan accordingly! You can find a SAMPLE PAPER in the content section of D2L.
Things to notice about the sample paper:
1) The overall approach of the paper is to avoid being bogged down in technical notation. It talks more about motivations and the main ideas while glossing over the “nitty-gritty” details.
2) Proofs are severely summarized. This is related to 1) above in that a real proof uses a lot of technical notation. Here the way of proof and the key properties used are mentioned without delving into specifics.
3) Theorems and the selected exercise are highlighted similar to a citation.
4) The sample exercise does not just appear randomly. It fits into the context of that portion of the paper.
5) Not all the examples of definitions and theorems are mentioned in the paper. Remember, this is a summary of the main points, so only important examples need to be mentioned and/or discussed.
6) The paper assumes the reader has perfect knowledge of previous chapters. Nowhere does it redefine terms or notation from a previous chapter. When you write your paper, you may assume the reader understands the previous chapters, especially those listed in the “Prerequisite Sections” in the table below.
7) In this specific case, Chapter 5 had a lot of theory and a few applications. As such, the paper mostly focuses on the new theory and only briefly mentions the applications. Make sure that your paper has the same ratio of theory to applications as the section you are summarizing.
Grading Rubric for
the Paper:
Expressing the main concept: 30 points
Abstraction/Statement of main definitions: 10 points
Statement of main theorems: 10 points
Abstraction of important proofs: 20 points
Appropriateness of example problem: 5 points
Explanation of solution to example problem: 15 points
Grammar and spelling: 10 points
----------------
100 points total
Paper Topics:
|
Topic |
Book Sections (summarize) |
Prerequisite Sections (assume reader knows) |
Approx. Date Material Covered |
Max Paper Length |
|
Symmetry and Counting |
Chapter 29 |
Chapter 5 |
9/24 |
3 pages |
|
Symmetry Groups, Frieze
Groups, |
Chapters 27 & 28 |
Chapters 5 & 7 |
10/8 |
3-4 pages |
|
Sylow Theorems |
Chapter 24 |
Chapters 7, 8, & 9 |
10/22 |
3 pages |
|
Generators & Relations |
Chapter 26 |
Chapters 8, 9, & 10 |
10/22 |
3 pages |
|
Finite Simple Groups |
Chapter 25 |
Chapters 5, 7, 9, & 11 |
10/22 |
3 pages |
|
Ring Homomorphisms |
Chapter 15 |
Chapters 10 & 14 |
11/9 |
3 pages |
|
Factorization of Polynomials |
Chapter 17 |
Chapters 14 & 16 |
11/9 |
3-4 pages |
|
Divisibility in Integral Domains |
Chapter 18 |
Chapter 14 & Def of PID (p. 297) |
11/9 |
3 pages |
|
Extension Fields |
Chapters 19 & 20 |
Chapters 14 & 16 |
11/9 |
3-4 pages |
Academic Dishonesty: Any type of academic dishonesty (cheating, copying, plagiarism, etc.) will result in failure and will be reported to school authorities. If you are having trouble with an assignment, please see the instructor first. The instructor reserves the right to “google” your paper to search for instances of plagiarism.
Note: This syllabus is subject to change if deemed necessary by the instructor.
Tentative Schedule of Events – Math 440
(subject to change)
|
Approx. Week |
Topics |
Book Sections |
Homework |
|
8/27 - 8/31 |
Introduction, Review, |
Chapters 0 - 2 |
pg 23: 3, 4, 15, 20, 28, 30, 37, 48, 53 pg 37: 11, 16 pg 53: 4, 5, 8, 12, 14, 26, 31, 33 |
|
9/3 - 9/7 |
Finite Groups & Subgroups |
Chapter 3 |
pg 67: 4, 6, 7, 9,
10, 12, 14, 15, 20, |
|
9/10 - 9/14 |
Cyclic Groups |
Chapter 4 |
pg 82: 4, 10, 11,
14, 19, 22, 24, 29, 30, 33, 50, 51, 63 Extra Credit: pg 90: 2, 8, 12, 16,
18, 34, 44 |
|
9/17 - 9/21 |
Permutation Groups |
Chapter 5 |
pg 112: 6, 8, 10,
13, 18, 22, 23, 26, 28, 31, |
|
9/24 - 9/28 |
Midterm |
Chapter 6 |
pg 132: 7, 9, 10, 14, 15, 18, 23, 27, 37, 43 (Due Wednesday 10/3) Take Home Exam Re-Do (Due Friday 10/5) |
|
10/1 - 10/5 |
Cosets & Lagrange's Theorem |
Chapters 7 &
9 Mon: pgs 137-140 Wed: pgs 140-143 Fri: pgs 177-179 |
pg 148: 4, 8, 11,
12, 14, 18, 21, 27, 35, 36, 47 (Due Wednesday 10/10) |
|
10/8 - 10/12 |
Normal Subgroups
& Factor Groups |
Chapters 9 &
10 Mon: pgs 179-200 Wed: pgs 203-209 |
pg 191: 10, 14, 18, 37, 39, 45, 56 pg 210: 2, 4, 6, 12, 25, 27 (Due Wednesday 10/17) |
|
10/15 - 10/19 |
Fundamental Theorem of Finite Abelian Groups Wednesday 10-11am: East Hall of Kryzsko Commons “Fenceposts and Euler's Theorem" See below for more info |
Chapters 8 &
11 Mon: pgs 153-157 Wed: pgs 217-222 Fri: Finish Groups |
pg 165: 7, 8, 9, 14, 15, 20, 25, 40, 41 pg 225: 2, 4, 10, 15, 19, 21 (Due Monday 10/22) Extra Credit: pg 174: 6, 14, 20, 32, 34 pg 230: 10, 14, 19, 32, 36 (Due Wednesday 10/31) |
|
10/22 - 10/26 |
Midterm Paper Q & A |
Mon: Review Wed: Exam II Chapter 12 Fri: pgs 235-238 |
No Homework. |
|
10/29 - 11/2 |
Integral Domains |
Chapters 12
& 13 Mon: pgs 238-240 Wed: pgs 248-251 Fri: pgs 252-254 |
pg 240: #2, 5, 18, 19, 22, 28, 29, 40 pg 254: #5, 8, 14, 23, 29, 38, 41a (Due Wednesday 11/7) |
|
11/5 - 11/9 |
Ideals and Factor Rings |
Chapter 14 Mon: pgs 261-262 Wed: pgs 262-265 Fri: pgs 266-268 |
pg 268: #5, 6, 8,
10, 12, 17, 18, 24, (Due Wednesday 11/14) |
|
11/12 - 11/16 |
Finite Fields |
Chapter 22 Mon: No Class Wed: pgs 381-385 (Quiz pgs 261-268) Fri: Computation Worksheet Take-Home Portion of Exam III |
Take-Home Portion of Exam III (Due Monday 11/19) Extra Credit: pg 275: #4, 6, 10, 20,
22, (Due Wednesday 11/28) |
|
11/19 |
Midterm |
Mon: In-Class Portion of Exam III Wed: No Class Fri: No Class |
No Homework. |
|
11/26 - 11/30 |
Elliptic Curves Paper Due on Friday |
Supplementary
Materials Mon: Elliptic Curves (Slides) Wed: Rational Elliptic Curves Fri: Computation Worksheet |
(Due Wednesday 12/5) |
|
12/3 - 12/7 |
Elliptic Curves over Finite Fields |
Supplementary
Materials Mon: Elliptic Curves over Finite Fields Wed: Computation Worksheet (Quiz/Activity) Fri: Semester Review |
Homework: Study for the final |
Final Exam
Wednesday, December 12
8am – 10am
GENERAL
ANNOUNCEMENTS
Distinguished Lectures in
Mathematics
By Dr. Manjul Bhargava
Professor, Mathematics
Department
Princeton University
Lecture I - Tuesday
10/16, 3:00 – 4:00 pm
Venue: East Hall, Kryzsko Commons
Title: The Mathematics of Secret Codes: From Ancient Times to the Modern Era
We will talk about various kinds of codes, originating in ancient cultures, and how they have evolved over the years mathematically and with regards to their practical use.
Lyceum Presentation
– Tuesday, 10/16, 7:30-8:50 pm
Venue: PAC Auditorium
Title: Linguistics, Drumming, Poetry and Mathematics
Abstract: Mathematics pervades all the sciences, but it also lies at the heart of a number of fields in the humanities. Two important such subjects, which go back to ancient times, are linguistics and music; in fact, many of the modern mathematical tools used in probability and combinatorics, and applied in varied technologies such as those on NASA space missions, originate in problems encountered by linguists and musicians thousands of years ago. A look at some of these ancient, poetic problems--and their remarkable solutions through the ages--reveals much about the nature of human thought and the origins of mathematics. The talk will include live demonstrations on the tabla, a traditional Indian percussion instrument.
Lecture II -
Wednesday 10/17, 10:00 – 11:00 am
Venue: East Hall, Kryzsko Commons
Title: Fenceposts and Euler's Theorem
Abstract: Many great theorems in mathematics (and physics!) can be proven by what are called "thought experiments". One beautiful and important such example from graph theory and solid geometry is what is known as "Euler's Theorem", which describes the possible numbers of sides, corners, and edges that a three-dimensional solid object (called a "polyhedron") can have. We will talk about how one can prove Euler's theorem through a thought experiment, by imagining what happens to a farmer and his fences when there is a big flood. We'll then apply this new knowledge to study these important and ubiquitous 3-dimensional objects known as "polyhedra". [A free stress-relieving polyhedron will be provided to all in attendance.]
Lecture III -
Wednesday 10/17, 3:00 – 4:00 pm
Venue: East Hall, Kryzsko Commons
Title: The representation of integers by quadratic forms
Abstract: The classical "Four Squares Theorem" of Lagrange asserts that any positive integer can be expressed as the sum of four squares; that is, the quadratic form a^2+b^2+c^2+d^2 represents all (positive) integers. When does a general quadratic form represent all integers? When does it represent all odd integers? When does it represent all primes? We show how all these questions turn out to have very simple and surprising answers. In particular, we will describe the recent progress (joint with J. Hanke, Duke University) in proving Conway's "290-Conjecture" for universal quadratic forms.
Commitment
to Inclusive Excellence: WSU recognizes that
our individual differences can deepen our understanding of one another and the
world around us, rather than divide us. In this class, people of all
ethnicities, genders, religions, ages, sexual orientations, disabilities,
socioeconomic backgrounds, regions, and nationalities are strongly encouraged
to share their rich array of perspectives and experiences. If you feel your differences may in some way
isolate you from WSU’s community or if you have a need of any specific
accommodations, please speak with the instructor early in the semester about
your concerns and what we can do together to help you become an active and
engaged member of our class and community.
Campus
Resources (Short version):
Campus
Resources (Long version):
The Standard Disclaimer
applies.