Abstract Algebra I
Math 311 (fall 2009)

Class meetings: Monday/Wednesday/Friday 10:00–10:50am, Gibson Hall 325

Instructor: Eric Rowland, erowland
Feel free to send me mail. Short questions work especially well. Longer questions tend to work better in office hours, but I will also try to answer them.

Office: Gibson Hall 421

Extra office hours: Sunday, December 6th, 2–4pm

Office hours: Tuesday 2:00–3:00pm, Friday 11–12pm, and by appointment

Text: Shifrin, Abstract Algebra: A Geometric Approach, Prentice Hall, 1996.
We will cover sections from chapters 1, 2, 3, 4, and 6. As an introduction to abstract algebra, this will take us through some number theory, ring theory, and group theory. If there is time we may also cover some of chapter 7.

About the course:
This course may be a little more interactive than other math courses you have taken. Our meetings will consist of a combination of discussion, lecture, and computer demonstrations. What this means is that meetings will not be primarily used for first exposure to new material, nor will we be able to cover everything in as much detail as the book. Therefore you are expected to read each section before we discuss it in class so that we can focus on intuition and understanding — the real mathematics.

Grading:
17.5% — exam 1
17.5% — exam 2
35% — final exam
20% — problem sets
10% — participation and project

Exams:
Exam 1 is on Friday, October 2nd and covers up through section 3.2.
Exam 2 is on Wednesday, November 4th and covers section 3.3 up through Lagrange's theorem and its corollaries in section 6.3.
The final exam is on Monday, December 7th, 8am–noon.

Projects:
Information about the projects for the course.

Problem sets:
Each Wednesday I will collect the exercises assigned the previous week. (The first assignment is an exception; see below.) I plan to grade some subset of the problems.
I'll try to assign a manageable number of problems each week. This is intended to let you focus on the problems that are assigned and enable you to do them well and write them up clearly.
The challenge problems are optional; however, they may be particularly interesting. They are a kind of extra credit in the sense that if your final grade is borderline, then having done a portion of the challenge problems will help.
You are strongly encouraged to work in groups, although the work you hand in must be your own presentation.
You are also strongly encouraged to use Mathematica as you read the text and as you work through the problems. Ask me for help when you need it; I enjoy answering Mathematica questions.
You may submit work to me electronically.

40. December 4th
[review]

39. December 2nd (not to be collected)
1. 6.4.14

38. November 30th
No problem set.

37. November 23rd (due Monday, November 30th)
Finish reading section 6.4.
1. 6.4.8 (Note that this is asking about conjugation in A5; these permutations are certainly conjugate in S5.)
2. 6.4.9

36. November 20th (due Monday, November 30th)
1. 6.4.1
2. 6.4.2(a,c,e)
3. 6.4.4

35. November 18th
Continue reading section 6.4.

34. November 16th
Start reading section 6.4 before Wednesday.
No problem set, since there don't seem to be any exercises explicitly about the fundamental homomorphism theorem. So just think about it. It might be worth looking back at the fundamental homomorphism theorem for rings and comparing the two.

33. November 13th (due November 18th)
1. 6.3.18
2. 6.3.19 (Note that this is another way to show that the subgroups of D4 of order 4 are normal.)

32. November 11th (due November 18th)
1. 6.3.4
2. 6.3.5
Challenge: 6.3.16

31. November 9th (due November 18th)
Finish reading section 6.3.
1. 6.1.2(c,d)
2. 6.2.6(b)
3. 6.2.9
4. 6.2.10

30. November 6th
No problem set.

29. November 4th
[exam 2]

28. November 2nd
No problem set. (This may be a good time to do some work on your project!)

27. October 30th (due November 4)
Finish reading section 6.3.
1. 6.3.1
2. 6.3.8
3. 6.3.13

26. October 28th (due November 4)
Finish reading section 6.2, and start reading section 6.3.
1. 6.2.2 (This notation is defined in Proposition 1.2.)
2. 6.2.5(b)
3. 6.2.8
4. 6.2.15(a,b,c)
Challenge: 6.2.12

25. October 26th (due November 4)
Start reading section 6.2.
1. 6.1.1 (I know there are a lot of parts here, but they're all good examples. Don't feel like you need to write long explanations.)
2. 6.1.2(a,b)
3. 6.1.5
4. 6.1.16

24. October 23rd (due October 28th)
Read section 6.1 before Monday.
1. 4.3.8 (Hint: It's true. It may help to become familiar with the proof of Proposition 3.5.)
2. 4.3.10(a,c)
3.
  (a) Make a conjecture about which primes can be written as p = a2 + 2 b2. You can modify the code from class to get started.
  (b) What about p = a2 + 3 b2?
  Challenge: What about p = a2 + 5 b2?
Challenge: 4.3.15

23. October 21st (due October 28th)
Finish reading section 4.3 before Friday.
1. 4.3.1
2. 4.3.2
3. 4.3.12 or 4.3.13

22. October 19th (due October 28th)
Read section 4.3 before Wednesday.
1. 4.2.27 (if you didn't do it last week)
Challenge: 4.2.24

21. October 14th (due October 21st)
1. four parts of 4.2.3
2. 4.2.11
Challenge: 4.2.21, 4.2.27

20. October 12th (due October 21st)
1. 4.1.12
2. two parts of 4.1.14

19. October 9th (due October 14th)
Finish reading section 4.2 before Monday.
1. 4.2.1
2. 4.2.2 (See page 127 for the definition of the direct product.)
3. 4.2.4
4. 4.2.8 (It may help to look at 4.1.11.)
Challenge: 4.2.20 (Dan's conjecture)

18. October 7th (due October 14th)
Read section 4.2 before Friday.
1. 4.1.4(b,c,d,f)
2. 4.1.6 (You may need to remember some things about binomial coefficients.)
3. 4.1.8
4. 4.1.13 (For each part, find some m such that the equation has no solution modulo m.)

17. October 5th (due October 14th)
Read section 4.1 before Wednesday if you haven't already.
1. 4.1.1
2. 4.1.2
3. 4.1.7

16. October 2nd
[exam 1]

15. September 30th (due October 7th)
Read section 4.1 before Monday.
1. five parts of 3.3.2
2. one part of 3.3.3
3. 3.3.5
Challenge: 3.3.9

14. September 25th (due September 30th)
Read section 3.3 before Wednesday.
1. three parts of 3.2.3
2. two parts of 3.2.6
3. 3.2.10
4. 3.2.16

13. September 23rd (due September 30th)
Read section 3.2 before Friday if you haven't already.
1. 3.1.2(a,b,c)
2. 3.1.8
3. three parts of 3.1.10
4. 3.2.2(a)
Challenge: 3.2.2(b,c)

12. September 21st (due September 30th)
Read section 3.2 before Wednesday.
1. 3.1.1(a) and 3.1.1(c)
2. 3.1.9
3. 3.1.14
4. Find an irreducible polynomial of degree 10 in Q[x]. (The Mathematica symbol Factor factors polynomials and therefore tells you when a polynomial is irreducible.)
Challenge: 3.1.19

11. September 18th (due September 23rd)
Read section 3.1 before Monday if you haven't already.
1. 2.5.10
2. 2.5.13
3. 2.5.15

10. September 16th (due September 23rd)
Read section 3.1 before Friday.
1. 2.5.2
2. 2.5.3
3. 2.5.5
4. 2.5.6

9. September 14th (due September 23rd)
Read section 2.5 before Wednesday.
1. 2.3.6(a)
2. 2.3.9(c)
3. 2.3.21
4. 2.3.22
5. Type "z^6 = 1" into Wolfram|Alpha (or use Mathematica) to plot the roots of z6 = 1 in the complex plane. Then plot the roots of z6 = –1 and z6 = i. Explain what these three pictures have in common and why.
Challenge: 2.3.11

8. September 11th (due September 16th)
Read section 2.4 before Monday.
1. 2.2.3 (that the arithmetic mean is not smaller than the geometric mean)
2. 2.2.4
3. 2.2.5
4. 2.2.12
Challenge: 2.2.6, 2.2.17

7. September 9th (due September 16th)
Read sections 2.2 and 2.3 before Friday.
1. 2.1.8
2. (part of 2.1.11) Why must R be an integral domain to construct the field of quotients of R?
3. 2.1.12 or 2.1.13
4. 2.1.14
Challenge: 2.1.16

6. September 4th (due September 9th)
Read section 2.1.
1. 1.4.1
2. 1.4.2
3. 1.4.3
4. 1.4.4
5. 1.4.6 or 1.4.7
6. 1.4.8
7. How many zero-divisors does the ring Z1000 have? Z999? Z998? Z997?
Challenge: 1.4.9, 1.4.13

5. September 2nd (due September 9th)
Here is the Mathematica notebook we created in class.
Read section 1.4 for Friday if you haven't already.
Please do this assignment in a Mathematica notebook and send it to me or print it, so I can see your code.
1. Let an be as in 1.1.19.
  (a) Make a picture (with Mathematica) of Pascal's triangle modulo 2, as the book suggests. The symbol ArrayPlot might be useful.
  (b) Compute an for n < 512 (or more if you can). Can you guess what the limit is?
2. Find the smallest example of a number of the form 3n + 1 that has two distinct mock-prime factorizations in the sense of 1.2.16(b).
3.
  (a) How many primes of the form 4k + 3 are there less than 10? 100? 1000? 10000? 105? 106?
  (b) Are there more primes of the form 4k + 1 or 4k + 3 in these ranges? Does this trend appear to continue? (How far can you go computationally?) This result is known as the Chebyshev bias.

4. August 31st (due September 9th)
1. four parts of 1.3.21
2. 1.3.22
3. 1.3.30: Start by finding all n less than 100 that cause the numerator n2 + 12 and denominator n + 4 to have a common divisor > 1. What is the pattern? Challenge: Why those numbers and not any others?

3. August 28th (due August 31st)
0. Read section 1.4.
1. 1.3.5
2&3. two of 1.3.14, 1.3.15, and 1.3.16
4. 1.3.20
5. Look at the last digit of the numbers 15, 25, 35, 45, 55, ... . What is the last digit of n5? Why?
Challenge: 1.3.28, 1.3.29

2. August 26th (due August 31st)
0. For Friday, look through Appendix A, and read any parts (especially A.3) that you feel you need to brush up on. Also, read section 1.3.
1. three parts of 1.2.1
2. 1.2.2
3. 1.2.5 or 1.2.6
4. 1.2.16(b)
5. How many primes are there less than 10? 100? 1000? 10000? 105? 106? (Ask me what some relevant Mathematica symbols are if you need some guidance.)
Challenge: 1.2.18

1. August 24th (due August 26th)
0. Download and install Mathematica from e-academy if you don't already have it.
1.
  (a) Use Mathematica to compute the largest prime factor of the coefficient of x177y223 in (x + y)400.
  (b) How could you have computed this without a computer?
2.
  (a) Come up with a Mathematica computation that you are quite certain would be infeasible without a computer, and run it.
  (b) Why do you think it would be infeasible without a computer?
3. Why are you taking Algebra I?
4. What do you expect to be doing with math in five or ten years?
0. Read sections 1.1 and 1.2 of the text.
5. one part (of your choosing) of exercise 1.1.4
6. exercise 1.1.11 (that each positive integer has a unique binary representation)
Challenge exercises: 1.1.8 and 1.1.19