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:
252: 14, 15
307: 12 – 14
309: 3, 4
350: 2 – 4
352: 2
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] .

Find the order of t in F_{9}.
Find a primitive element of F_{9}.
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
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:
248: 7 – 9, 12
251: 4, 5, 7
Why should 3 and 31 have the same order mod 7 ?
Show that 3 is primitive in Z/49Z.
Show that 31 is not primitive in Z/49Z.
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:
 Read: § 15C
Exercises:
243: 10, 11
246: 1, 2, 4, 6
Show that 2 is a primitive element in Z/mZ for
m = 11 and 19 but not for m = 11 * 19 = 209.
Find all roots of the polynomial x^{3}  1 in (Z/mZ)[t] for
m = 11, 19, and 209.
What is the largest order of any element in Z/209Z ?
 Fri., Nov. 2:
 Read: § 15B
Exercises:
243: 4, 5, 7, 8
245: 1
Find primitive elements for Z/17Z,
Z/23Z, and Z/34Z.
Show that there is no primitive element in Z/32Z.
 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(iiii)
 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:
40487/257
(1 + SQRT{5})/2
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
61/67.
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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