Classical Algebra
Math 326
Assignments

Fall Semester, 2007

Assignments are listed by the date due. A PDF version of this page is available for printing.

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.
Bring your questions.

Wed., Dec. 5:

Quiz Read: § 28C
Exercises:

418: 8, 11, 12

421: 5 – 7, 10, 11

425: 4 – 6, 8 – 12

Mon., Dec. 3:

Read: §§ 28A, 28B
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:

Read: §§ 24A, 24B
Exercises:

350: 7, 9 – 11

352: 7

356: 2 – 6

361: 5

Wed., Nov. 28:

Read: §§ 23A, 23B
Exercises:

1. 252: 14, 15

2. 307: 12 – 14

3. 309: 3, 4

4. 350: 2 – 4

5. 352: 2

6. The field ${\mathbf{F}}_9$ of $9$ elements is defined as a ring of congruence classes: $${\mathbf{F}}_9={\mathbf{F}}_3\left[t\right]⁄\left(t^2+1\right){\mathbf{F}}_3\left[t\right]\text{.}$$

   1. Find the order of $t$ in ${\mathbf{F}}_9$.

   2. Find a primitive element of ${\mathbf{F}}_9$.

   3. How many primitive elements does ${\mathbf{F}}_9$ 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
Read: § 20B
Exercises:

262: 9 – 11

304: 4 – 7

307: 8 – 11

309: 1, 2

Fri., Nov. 16:

Quiz
Read: § 20A
Exercises:

251: 12, 13, 16

258: 7

262: 5 – 8

303: 1 – 3

Wed., Nov. 14:

Read: §§ 16C, 16D
Exercises:

249: 14, 15

251: 10, 11

257: 4 – 6

262: 1 – 4

Mon., Nov. 12:

Quiz
Read: §§ 16A, 16B
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:

Read: § 15D
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 $\mathbf{Z}⁄49\mathbf{Z}$.

5. Show that $31$ is not primitive in $\mathbf{Z}⁄49\mathbf{Z}$.

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

Mon., Nov. 5:

Read: § 15C
Exercises:

1. 243: 10, 11

2. 246: 1, 2, 4, 6

3. Show that $2$ is a primitive element in $\mathbf{Z}⁄m\mathbf{Z}$ for $m=11\text{and}19$ but not for $m=11*19=209$.

4. Find all roots of the polynomial $x^3-1$ in $\left(\mathbf{Z}⁄m\mathbf{Z}\right)\left[t\right]$ for $m=11$, $19$, and $209$.

5. What is the largest order of any element in $\mathbf{Z}⁄209\mathbf{Z}$ ?

Fri., Nov. 2:

Read: § 15B
Exercises:

1. 243: 4, 5, 7, 8

2. 245: 1

3. Find primitive elements for $\mathbf{Z}⁄17\mathbf{Z}$, $\mathbf{Z}⁄23\mathbf{Z}$, and $\mathbf{Z}⁄34\mathbf{Z}$.

4. Show that there is no primitive element in $\mathbf{Z}⁄32\mathbf{Z}$.

Wed., Oct. 31:

Read: § 15A
Exercises:

163: 22

200: 13, 15

205: 6, 8

236: 5

238: 7 – 9

241: 1

243: 6

Mon., Oct. 29:

Read: Ch. 14
Exercises:

163: 18, 21

200: 11, 12

205: 5, 7

233: 1, 2

234: 3

236: 4

Fri., Oct. 26:

Read: § 12B
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:

Read: § 12A
Exercises:

141: 18

163: 10, 11, 13, 14

169: 3, 4

196: 1 – 3

Mon., Oct. 15:

Read: § 10B
Exercises:

141: 17, 19

145: 16, 17

147: 6

153: 3, 8

158: 3, 4, 7

166: 1, 2

Fri., Oct. 12:

Quiz
Read: §§ 9F, 10A
Exercises:

138: 14 – 18

141: 13, 15, 16

145: 12, 13, 15

147: 3, 5

153: 1, 2

158: 1, 2

Wed., Oct. 10:

Read: §§ 9D, 9E
Exercises:

122: 16, 17

137: 10 – 13

141: 6, 9 – 11, 14

144: 8, 11(i-iii)

147: 2, 4

Mon., Oct. 8:

Read: §§ 9B, 9C
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
Read: §§ 8C, 9A
Exercises:

90: 12, 13

122: 9, 12, 13

125: 5 – 7

133: 5, 6, 9

136: 1, 2, 4

Mon., Oct. 1:

Read: §§ 8A, 8B
Exercises:

74: 9

86: 8, 9

89: 4 – 7, 9 – 11

121: 1, 3, 5, 8

125: 2 – 4

Fri., Sep. 28:

Read: §§ 6D, 6E
Exercises:

73: 6 – 8

84: 2 – 5

86: 1 – 3, 5, 7

89: 1 – 3

Wed., Sep. 26:

Quiz
Read: §§ 6A, 6B, 6C
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:

Read: §§ 5D, 5E
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
Read: §§ 5B, 5C
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:

Read: §§ 4C, 5A
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
Read: §§ 4A, 4B
Exercises:

35: 8, 9, 13, 20

45: 3

49: 1, 2

51: 1, 2

Mon., Sep. 10:

Read: § 3E
Exercises:

33: 4(iv, v)

35: 6, 7(i, ii)

45: 1

Find the continued fraction expansion of:

1. $40487⁄257$

2. $\left(1+\sqrt{5}\right)⁄2$

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 $\left[3,2,7,2\right]$.

Find the continued fraction expansions of

1. $61⁄67$.

2. $44⁄37$.

Wed., Sep. 5:

Read: §§ 3B, 3C
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.

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