Classical Algebra Math 326 Assignments

Fall Semester, 2007

These are exercises designed to prepare you for the quizzes and the written assignments. Those which are to be submitted as written assignments are so labeled. While you may find it helpful to discuss the exercises with others, no collaboration is permitted on the written assignments.

Tue., Dec. 11:
Final Examination: 3:30 – 5:30
Mon., Dec. 10:
Office hours: 2:15 – 4:15
Fri., Dec. 7:
Written Assignment No. 5 (also available as PDF) is due.
Wed., Dec. 5:
Exercises:
418: 8, 11, 12
421: 5 – 7, 10, 11
425: 4 – 6, 8 – 12
Mon., Dec. 3:
Exercises:
351: 12, 13, 15
356: 7, 10
416: 2, 3, 5, 6
421: 1, 2, 4
Find a primitive root mod 343.
Fri., Nov. 30:
Exercises:
350: 7, 9 – 11
352: 7
356: 2 – 6
361: 5
Wed., Nov. 28:
Exercises:
1. 252: 14, 15

2. 307: 12 – 14

3. 309: 3, 4

4. 350: 2 – 4

5. 352: 2

6. The field F_{9} of 9 elements is defined as a ring of congruence classes:

 F_{9}  =  F_{3}[t]/(t^{2} + 1)F_{3}[t]    .

1. Find the order of t in F_{9}.

2. Find a primitive element of F_{9}.

3. How many primitive elements does F_{9} have?

Mon., Nov. 26:
Quiz
Class will be held, but there is no specific assignment for this day.
Wed–Fri., Nov. 21–23:
University Recess
Mon., Nov. 19:
Quiz
Exercises:
262: 9 – 11
304: 4 – 7
307: 8 – 11
309: 1, 2
Fri., Nov. 16:
Quiz
Exercises:
251: 12, 13, 16
258: 7
262: 5 – 8
303: 1 – 3
Wed., Nov. 14:
Exercises:
249: 14, 15
251: 10, 11
257: 4 – 6
262: 1 – 4
Mon., Nov. 12:
Quiz
Exercises:
249: 10, 11
251: 8, 9
257: 1 – 3
Fri., Nov. 9:
Written Assignment No. 4 (also available as PDF) is due.
Wed., Nov. 7:
Exercises:
1. 248: 7 – 9, 12

2. 251: 4, 5, 7

3. Why should 3 and 31 have the same order mod 7 ?

4. Show that 3 is primitive in Z/49Z.

5. Show that 31 is not primitive in Z/49Z.

6. Find the smallest prime p > 2 for which the smallest positive integer c that is primitive in Z/pZ is not also primitive in Z/p^{2}Z.

Mon., Nov. 5:
Exercises:
1. 243: 10, 11

2. 246: 1, 2, 4, 6

3. Show that 2 is a primitive element in Z/mZ for m = 11 and 19 but not for m = 11 * 19 = 209.

4. Find all roots of the polynomial x^{3} - 1 in (Z/mZ)[t] for m = 11, 19, and 209.

5. What is the largest order of any element in Z/209Z ?

Fri., Nov. 2:
Exercises:
1. 243: 4, 5, 7, 8

2. 245: 1

3. Find primitive elements for Z/17Z, Z/23Z, and Z/34Z.

4. Show that there is no primitive element in Z/32Z.

Wed., Oct. 31:
Exercises:
163: 22
200: 13, 15
205: 6, 8
236: 5
238: 7 – 9
241: 1
243: 6
Mon., Oct. 29:
Exercises:
163: 18, 21
200: 11, 12
205: 5, 7
233: 1, 2
234: 3
236: 4
Fri., Oct. 26:
Exercises:
163: 15 – 17
169: 5, 6
196: 5 – 7, 10
203: 1, 3
Wed., Oct. 24:
Midterm Test in class
Mon., Oct. 22:
Review Session: bring questions
Fri., Oct. 19:
Written Assignment No. 3 (also available as PDF) is due.
Wed., Oct. 17:
Exercises:
141: 18
163: 10, 11, 13, 14
169: 3, 4
196: 1 – 3
Mon., Oct. 15:
Exercises:
141: 17, 19
145: 16, 17
147: 6
153: 3, 8
158: 3, 4, 7
166: 1, 2
Fri., Oct. 12:
Quiz
Exercises:
138: 14 – 18
141: 13, 15, 16
145: 12, 13, 15
147: 3, 5
153: 1, 2
158: 1, 2
Wed., Oct. 10:
Exercises:
122: 16, 17
137: 10 – 13
141: 6, 9 – 11, 14
144: 8, 11(i-iii)
147: 2, 4
Mon., Oct. 8:
Exercises:
122: 14, 15
127: 8, 9
133: 7, 10
137: 3, 5, 7 – 9
141: 3 – 5, 7
142: 1
Fri., Oct. 5:
Written Assignment No. 2 (also available as PDF) is due.
(The written assignments must be submitted on paper – no email.)
Wed., Oct. 3:
Quiz
Exercises:
90: 12, 13
122: 9, 12, 13
125: 5 – 7
133: 5, 6, 9
136: 1, 2, 4
Mon., Oct. 1:
Exercises:
74: 9
86: 8, 9
89: 4 – 7, 9 – 11
121: 1, 3, 5, 8
125: 2 – 4
Fri., Sep. 28:
Exercises:
73: 6 – 8
84: 2 – 5
86: 1 – 3, 5, 7
89: 1 – 3
Wed., Sep. 26:
Quiz
Exercises:
54: 32 – 34
62: 1, 2
67: 9
70: 6
72: 2, 4, 5
73: 2 – 5
80: 3
84: 1

Problem solution (also available as PDF) requested by not presented during Monday's class.

Mon., Sep. 24:
Exercises:
52: 15, 16
54: 28, 30, 31
59: 5
65: 7, 8
67: 7, 8
70: 4, 5
72: 1, 3
73: 1
Fri., Sep. 21:
Written Assignment No. 1 (also available as PDF) is due.
Wed., Sep. 19:
QUIZ
Exercises:
35: 20
51: 10, 11, 13
53: 19 – 21, 25
65: 5, 6
67: 2, 3, 5, 6
70: 1, 3
Mon., Sep. 17:
Exercises:
35: 11, 14, 18
50: 3, 4
51: 3, 4, 7 – 9
64: 1, 3, 4
Fri., Sep. 14:
University in Recess: no class
Wed., Sep. 12:
QUIZ
Exercises:
35: 8, 9, 13, 20
45: 3
49: 1, 2
51: 1, 2
Mon., Sep. 10:
Exercises:
33: 4(iv, v)
35: 6, 7(i, ii)
45: 1
Find the continued fraction expansion of:
1. 40487/257

2. (1 + SQRT{5})/2

3. SQRT{17}

Fri., Sep. 7:
Quiz
Read: the course notes on Continued Fractions (also available as PDF)
Exercises:
29: 4, 6(ii, iv)
33: 2, 3, 4(iii)
35: 5
Evaluate the continued fraction represented by the sequence [3,2,7,2] .
Find the continued fraction expansions of
1. 61/67.

2. 44/37.

Wed., Sep. 5:
Exercises:
23: 5
24: 5
27: 6, 9, 10, 11
29: 2, 5, 6(iii, iv)
33: 4(i)
And this: Recall that 20314_{5} was found to be 1334_{10}. Cipher in base 5 to convert this number from base 5 to base 7, and then check that result by converting it from base 7 to base 10.

Announcement: Undergraduates in the Department are invited to join the math club. There is a Yahoo online group called “albanymath” that students can join for sharing ideas and posting and receiving announcements of events and resources.

Mon., Sep. 3:
Labor Day Recess: no class
Fri., Aug. 31:
Read: §§ 2D – 2F, 3A, 3B
Exercises:
6: 4
11: 6, 8
18: 2
19: 1
23: 1 – 4
24: 1 – 3
27: 5
Wed., Aug. 29:
Read: §§ 1, 2A – 2C
Exercises:
6: 2
11: 2, 4, 5, 10, 14
15: 2, 4
If you wish to enter the writing intensive division of the course (Math 326Z), please be sure to submit the required essay (also available as PDF) at this class (or earlier).
Mon., Aug. 27:
First meeting: no assignment.