Classical Algebra
Math 326
Assignments

Fall Semester, 2008

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

Most of these assignments consist of practice 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 practice exercises with others, no collaboration is permitted on the problems in the written assignments.

Fri., Dec. 12:

Final Examination, 10:30 – 12:30, in ES 143

Exam Period Office Hours

Wed.	Dec. 10	2:30 – 4:00
Thurs.	Dec. 11	2:30 – 4:00

Fri., Dec. 5:

Handout: Solutions for Written Assignment No. 5

Thu., Dec. 4:

Last regular class meeting:
• Written Assignment No. 5 (also available as PDF) is due.
• Bring questions for review.

Tue., Dec. 2:

Expect a quiz.
Read: §§ 28B, 28C
Exercises:

351: 13, 15

356: 6, 7, 10

418: 8, 11, 12

421: 1, 2, 5 – 7

425: 4 – 6, 9 – 12

Tue., Nov. 25:

Read: §§ 24A, 28A
Exercises:

252: 14, 15

263: 11

307: 8 – 10, 12, 14

350: 9, 10, 12

356: 2 – 5

416: 2, 3, 5, 6

• Recall that rational numbers, i.e., integer ratios, have decimal expansions relative to a given base. Find the analogous expansion for the ratio of polynomials with coefficients in ${\mathbf{F}}_5$ $$\frac{1}{x-2}$$ relative to $\left(x+1\right)$ as the polynomial base. Do the “digits” repeat for this example?

Thu., Nov. 20:

Expect a quiz.
Read: §§ 20A, 20B, 23A
Exercises:

251: 10 – 13

257: 3, 5, 6

262: 7, 9, 10

303: 1 – 5

309: 1 – 3

350: 2 – 4, 11

Tue., Nov. 18:

Written Assignment No. 4 (also available as PDF) is due.

Thu., Nov. 13:

Read: §§ 16A, 16B, 16C, 16D
Exercises:

243: 10, 11

249: 12, 13

252: 11 – 13

257: 1, 2

262: 1 – 3, 6

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

• 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 $209$.

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

Tues., Nov. 11:

Read: §§ 15C, 15D
Exercises:

205: 7, 8

243: 6, 7, 8

246: 1 – 4, 6

248: 6 – 10

251: 7 – 9

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

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

Thu., Nov. 6:

Read: §§ 15A, 15B
Exercises:

163: 18, 21, 22

200: 13, 15

205: 5, 6

236: 4, 5

238: 7 – 9

241: 1

243: 4, 5

245: 1

Tue., Nov. 4:

Expect a quiz.
Read: §§ 12B, 14
Exercises:

141: 18

163: 15 – 17

200: 5 – 7, 11, 12

202: 1, 3

233: 1 – 3

Thu., Oct. 30:

Written Assignment No. 3 (also available as PDF) is due.

Tue., Oct. 28:

Read: §§ 10A, 10B, 12A
Exercises:

141: 16, 17, 19

145: 16, 17

163: 10, 11, 13, 14

166: 1, 2

169: 3, 4

196: 1 – 3

Thu., Oct. 23:

Midterm Test in class

Tue., Oct. 21:

Bring review questions
Read: § 9D
Exercises:

137: 7 – 9

142: 1, 4

144: 8

147: 2, 4, 6(i)

Thu., Oct. 16:

Expect a quiz.
Read: §§ 9A, 9B, 9C
Exercises:

110: 5(a) with $x_0=15$, 5(b) with $x_0=36$

123: 14, 15

133: 5 – 10

136: 1 – 5

141: 3 – 5, 7

Tue., Oct. 14:

Read: §§ 8A, 8B, 8C
Exercises:

89: 6, 7, 9 – 13

102: 3

105: 2

110: 4(c, d, e, f)

121: 1 – 5, 8, 9, 12, 13

125: 2 – 9

Thu., Oct. 9:

University recess: no class

Tue., Oct. 7:

Written Assignment No. 2 (also available as PDF) is due

Thu., Oct. 2:

Read: §§ 6E, 7C, 7D, 7E
Exercises:

74: 9

86: 5, 7 – 9

89: 1 – 5

100: 1 (i – iv), 2

105: 1, 3

110: 4(a, b)

Tue., Sep. 30:

University recess: no class

Thu., Sep. 25:

Read: §§ 6A, 6B, 6C, 6D
Exercises:

54: 32 – 34

67: 9

70: 6

72: 4, 5

73: 2 – 8

80: 3

84: 1 – 5

86: 1 – 4

Tue., Sep. 23:

Read: §§ 5C, 5D, 5E
Exercises:

52: 10, 11, 13, 15

54: 28, 30, 31

65: 6 – 8

67: 2, 3, 5 – 8

70: 1 – 5

72: 1 – 3

73: 1

Thu., Sep. 18:

Read: §§ 4C, 5A, 5B
Exercises:

35: 18

50: 3, 4

51: 3 – 5, 7 – 9

53: 19 – 21, 25

55: 1

64: 1, 3, 4, 5

Tue., Sep. 16:

Written Assignment No. 1 (also available as PDF) is due

Thu., Sep. 11:

Read: §§ 3E, 4A, 4B
Exercises:

1. 35: 8, 9, 13, 20

2. 45: 1, 3

3. 49: 1, 2

4. 51: 1, 2

5. Find the continued fraction expansion of:

   1. $40487⁄257$.

   2. $\frac{1+\sqrt{5}}{2}$.

   3. $\sqrt{17}$.

Tue., Sep. 9:

Read: Read the notes on Continued Fractions
Exercises:

1. 29: 4, 6(ii, v)

2. 33: 2, 3, 4(iii, iv, v)

3. 35: 5, 6, 7(i, ii)

4. And these:

   1. Find the continued fraction expansions of the rational numbers

      1. $\frac{61}{67}$.

      2. $\frac{44}{37}$.

   2. Find the continued fraction expansion of $\sqrt{2}$.

   3. Find the first three integers in the continued fraction expansion of $\sqrt[3]{5}$. (Feel free to use a calculator for this one.)

   4. Evaluate the finite continued fraction represented by the sequence $\left[3,2,7,2\right]$.

Thu., Sep. 4:

Expect a quiz
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 $10$.

Supplementary course notes on Induction (also available as PDF) are available here.

Tue., Sep. 2:

Read: §§ 2D – 2F, 3A, 3B
Exercises:

6: 4

11: 6, 8

18: 2

19: 1

23: 1 – 4

24: 1 – 3

27: 5

Thu., Aug. 28:

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).

Tue., Aug. 26:

First meeting: no assignment.

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