Math 336 Assignments

General remarks

1. Homework should be legible and neat, and pages should be fastened with a staple or paperclip.

2. Homework will be collected on Fridays and will cover the material through Wednesday's lecture (roughly). You should start attempting the problems as soon as possible after the material is presented in lecture. Please don't save it all for Thursday night.

3. Solutions should be written carefully, using good English, complete sentences, and adequate detail. Some of these solutions will be proofs. A good guideline here is that you should write proofs the way you would like to see them in your textbook.

4. Some of the problems have hints (not necessarily complete solutions) in the back of the book. If you get stuck on a problem, you might find it helpful to look at a nearby unassigned problem that has a hint.

Assignment 1, due Friday, January 25

Section 2A, p. 11: 1, 2
Section 3B, p. 29: 6(i)
Section 3C, pp. 33-36: 4(i), 5, 6(i), 7(i), 8, 20, 21 (extra credit)

Additional problems: 1, 2(a), 2(b) (e.c.)

(If your browser doesn't display the additional problems when you click on the link above, try right-clicking and selecting "save target as..." to download the file to your computer. See the announcement about this on the course home page for more details.)

Assignment 2, due Friday, February 1

Section 4A, p 50: 3
Section 4B, pp. 51-54: 2, 15, 31
Section 5A, p. 65: 5, 6, 7
Section 5B, p. 67: 2, 5, 6, 7, 8

Additional problems: 3, 4 (e.c.)

Assignment 3, due Friday, February 8

Section 5D, p. 72: 2, 3, 7
Section 5E, pp. 73-74: 1, 4, 7
Section 6A, p. 80: 2, 3
Section 6C, p. 84: 2, 3

Additional problems: 5, 6

Assignment 4, due Friday, February 15

6D, p. 86: 3, 4, 8
6E, p. 89: 5, 8(i), 12
8A, pp. 121-124: 3, 10, 17, 18
8B, pp. 125-126: 5, 7 [There is a typo in problem 7; you should find the order of 1+i.]
CS2.4, p. 8: 1, 2, 4, 7a,b

Extra-credit problem: Find an error in the first part of the proof of Proposition 2 on p. 123 of Childs, and correct that part of the proof.

Prelim 1 is on Thursday, February 21.

Assignment 5, due Monday, February 25

CS2.4: 6, 10 (e.c.) [Note: The code should be assumed binary in problem 10.]
8C, p. 133: 5, 7, 11
9A, pp. 136-138: 2, 8 (e.c.), 16
9B, p. 141: 3, 12, 19
9C, pp. 144-145: 7, 10 (e.c.), 15

Additional problems: 8, 9

Assignment 6, due Friday, March 1

9E, p. 151: 6
10B, p. 169: 6 [You don't need to read 10B in order to do this problem.]

Additional problems: 10, 11, 12, 13

Note: This assignment is intentionally short, but it's not as short as it looks; additional problem 13 might be time consuming. If you want to use a computer to save some of the work (such as computing powers mod m), that's fine as long as you write the program yourself and turn in the source code.

Assignment 7, due Friday, March 8

11A, p. 182: 1 [The hint in the back of the book is misleading], 2, 4, 5
11B, p. 185: 1(i)
CS3.4, pp. 17-18: 3, 4, 5, 7

Additional problems: 14a, 14b (e.c.), 14c, 15 (e.c.)

Assignment 8, due Friday, March 15

CS3.4, pp. 17-18: 9, 10, 11, 12, 13
11D, p. 188: 2, 3

Extra credit problem: Read the New York Times article about the hat problem. Consider the version of the game with 7 players, and devise a strategy that wins with probability 7/8. [Hint: Use the fact that the Hamming [7,4] code is perfect and that a random 7-bit binary word is unlikely to be a code word.]

Assignment 9, due Friday, March 29

12A, pp. 196-200: 1, 9(ii), 12, 15 [You can use any method on these problems.]
12B, pp. 202-205: 1, 5, 6(i) [Ignore the reference to Section 7B, E3.]
12C, pp. 206-207: 4, 6, 8 [If you have trouble with 6, take a look at 5.]
14, pp. 233-236: 1, 3, 5(b)

Prelim 2 is on Thursday, April 4.

Assignment 10, due Monday, April 8

15A, p. 243: 5, 11(ii), 12. The hint in the back of the book for E5(ii) doesn't give the complete answer. It takes some care to do this right.
15C, p. 249: 8(iii), 9, 12
15D, p. 251-252: 10, 12, 13(vi)
20A, p. 307: 9(i), 14(iii)
20B, p. 309: 2(ii). The modulus should be x⁴ + x + 1, not x⁴ + x + x.

Assignment 11, due Friday, April 12

21A, pp. 311-313: 1, 3, 5. The index  i  in problem 5 should range from 0 to d, not 1 to d.
23A, p. 350: 3, 5, 9
Addtional problem: Do the exercise at the end of the primitive root handout.

Assignment 12, due Friday, April 19

28A, pp. 416-418: 3, 5(ii), 8(iii), 13(e.c.) [Hint: See 23A, E9, p. 350.]
28B, p. 421: 2, 6, 7a(ii)
28C, pp. 425-426: 2, 3, 7, 12, 21. The notation in E12 may confuse you; F is a field with 4 elements, and F[t] is the ring of polynomials over F in one variable t. In E21 you may use the fact that F necessarily has characteristic 3.
28D, p. 428: 5. Note that F₉ here is the same F₉ that occurred in Exercise 7 on p. 126.
Additional problem: 16 (e.c.)

Assignment 13, due Friday, April 26

CS4.6, pp. 27-28: 1, 2, 3, 4, 5, 7, 10, 11. In problem 7 you can simply list coset representatives; give them in the form of polynomials. To make the problem more interesting/challenging, see if you can find a way to describe all the coset representatives in one sentence. Suggestion for problem 10: Do additional problem 17 first.

Additional problem: 17

Assignment 14, due Friday, May 3

11E, p. 193: 1, 2, 3, 4
CS3.4, p. 17: 8. [The problem is misplaced; it belongs in Chapter 4.]
CS4.6, p. 28: 15, 16, 17 (e.c.)

Additional problems: 18, 19, 20, 21, 22

Note: Problems 1, 2, and 4 on p. 193 are poorly phrased. In problems 1 and 2, the author wants you to give the homomorphism provided by the proof of Cayley's theorem. In problem 1, for instance, you should list the elements of U₈ and, for each element  a, describe explicitly the permutation L_a of U₈ given in the proof of Cayley's theorem. Your explicit description can look like those on p. 189, or you can use any other unambiguous way of describing a permutation. Problem 4 is similar; the author says in his hint what he has in mind.

Assignment 15, due never

Additional problems: 23, 24