Math 3360 Assignments, Spring 2009

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.


5. Assigned problems may change over the term, so check the assignment each week before you do it.

Assignment 1, due Friday, January 23

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

Assignment 2, due Friday,  January 30

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

Assignment 3, due Friday, February 6

Section 5D, p. 72: 2, 3, 7
Section 5E, pp. 73-74: 2, 7, 8
Section 6A, p. 80: 2, 3
Section 6C, p. 84: 2, 3
6D, p. 86: 3, 4, 8
6E, p. 89: 5, 8(i), 12

Assignment 4, due Friday, February 13

8A, pp. 121-124: 3, 8, 10, 17, 18
8B, pp. 125-126: 5, 7 [There is a typo in problem 7; you should find the order of 1+i.]
8C, p. 133: 5, 7, 11
9A, pp. 136-138: 2, 16
9B, p. 141: 3, 12, 19

Prelim 1 is on Tuesday, February 17.

Assignment 5, due Friday, February 27

9C, pp. 144-145: 7, 15
9E, p. 152: 7
10B, p. 169: 6 [You don't need to read 10B in order to do this problem.]
11A, p. 182: 1, 2, 4, 5
11B, p. 185: 1(i)

Additionally, the following two problems:

1. Using p = 19, q = 23, and encoding exponent e = 5, encode your initials. Use a = 11, b = 12, . . . , z = 36.

2. Consider an RSA cryptosystem with modulus 7081 and encoding exponent 1789. Using the same alphabet as in the previous problem, decode 5192, 2604, 4222. Show all your work. [Hint: 7081 is a product of two primes that are about the same size.]

Note: This last problem 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 6, due Friday, March 6

11D, p. 188: 2, 3
11E, p. 193: 1, 2, 3, 4
12A, pp. 196-200: 3, 9(ii), 12, 15 [You can use any method on these problems.]
12B, pp. 202-205: 1, 5, 6(ii) [Ignore the reference to Section 7B, E3.]

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 7, due Friday, March 13

CS sect. 1.6, p. 16: 1.3, 1.7
CS sect. 2.9, pp. 36-38: 2.5, 2.6, 2.12, 2.18

Assignment 8, due Friday, March 27

CS sect. 2.9, pp. 36-38: 2.19
14, pp. 233-236: 1, 2, 3
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

Assignment 9, due Friday, April 3

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

Prelim 2 is on Thursday, April 9.

Assignment 10, due Friday, April 17

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.

28A, pp. 416-418: 3, 5(ii), 8(iii), 13 [Hint: See 23A, E9, p. 350.]

Assignment 11, due Friday, April 24

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.

Assignment 12, due Friday, May 1

CS sect 3.6, pp. 49-50: 3.1, 3.2, 3.3, 3.5, 3.7, 3.8
CS sect 4.4, p. 58: 4.1, 4.5

Assignment 13, due never

Problem:

• Construct a field F₁₆ with 16 elements, and find a primitive element (i.e., an element of order 15).
  
• Group the elements of F₁₆ into conjugacy classes with respect to F₂ .
  
• Use your answer to (b) to say how many irreducible factors the polynomial x¹⁵ - 1 has in F₂ [x] and what their degrees are.