MATH 5331 HOMEWORK (Spring 2022)

MATH 5331 HOMEWORK (Spring 2022)

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Book =

Abstract Algebra, by D. S. Dummit & R. M. Foote, 3rd Ed., Wiley
A list of known errata for this book is posted here.

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Homework will not be collected.
There might be more than one correct answer for any given question.
Dates indicate homework assigned in lecture on that date. Dates for future assignments are tentative, and subject to change.
If you notice some questions are in a nonincreasing order, then it means that the order is recommended by your instructor and is usually chosen to match the order in which the material was presented in class.
Any question of the form Hi (i = 1,….) is designed to help you answer some textbook questions – make sure you answer any Hi question before working on a textbook problem that is listed after it.
Any optional exercises are for those students who wish to explore some of the ideas in more depth – those questions are NOT game for the tests.
Skimming through the main ideas in a section shortly before that section is covered in class should help you understand the lecture -- try it!
Recall that our test dates are: Feb 10, Mar 10, April 7, May 5.
LAST REVISION: 04/12/22.

Jan 18	Attendance will be noted, starting today. Read course syllabus carefully. Make a note of the test dates in your calendar. Review course website and repeat frequently during the semester. Read these study techniques and https://www.jeffreybennett.com/pdf/How_to_Succeed_general.pdf for ideas on how to study most effectively. Watch (or possibly rewatch) the video for Jan 18 on Canvas BBB page. Note that 1 or 2 minutes were added at the end to include a groupme link (created by a student in the class) in the recording’s chat and that link can also be found on the Canvas chat page. This video can also be found on the Canvas Modules page. Read pgs 1-3 & do pg 4: 1, 5. Read pgs 4-7 & do pgs 7-8: 1(a), 7. Read pgs 8-11 & do pgs 11-12: 3, 9, 15(a).
Jan 20	Watch (or possibly rewatch) the video for Jan 20 on Canvas BBB page; this video can also be found on the Canvas Modules page. Read your lecture notes & pgs 16-21 & do pgs 21-23: 6(a)-(c), 8, 9. Note the notation discussion on pgs 19-20 and do pgs 21-23: 25-27, 29 and H1.
Jan 25	Watch (or possibly rewatch) the video for Jan 25 on Canvas BBB page; this video can also be found on the Canvas Modules page. Read your lecture notes & pgs 16-21 & do pgs 21-23: 12, 22, 30 (hint: consider division algorithm on pg 4 of book). Read pgs 23-27 & pgs 29-32 & do pg 28: 14, 16 and pgs 32-33: 1, 4(a). Read pg 34 & pg 36 and do pg 35: 2, 3, 10, 11(a)(b) and pg 36: 1. By this time, you have seen quite a few theorems. It is best NOT to memorize the theorems, but to do enough of the homework so that the results of the theorems become known to you, though perhaps with different wording or using pictures.
Jan 27	Watch (or possibly rewatch) the video for Jan 27 on Canvas BBB page; this video can also be found on the Canvas Modules page. Note that this recording contains an extra 33 minutes at the end that was not covered in lecture. Read your lecture notes & pgs 36-39 & do pgs 39-41: 1, 13, 14, 15, 18, 20, 26. Read pgs 41-44 & do pgs 44-45: 2, 16, 17,, 4, 6.
Feb 01	Watch (or possibly rewatch) the video for Feb 1 on Canvas BBB page; this video can also be found on the Canvas Modules page. Read your lecture notes & pgs 46-48 & do pgs 48-49: 1(a)(e), 2(d), 10, 11(a)(c), 15, & do H2. Read pgs 51-52 & do pgs 52-53: 2, 3, 10, 11, 12(a)-(c).
Feb 03	LECTURE CANCELLED BY ORDER OF UTA (inclement weather). (See e-mail sent Feb 2.) Lectures resume on Tuesday, on campus, in PKH 109. Test 1 will be 1 week from today; see Canvas for an information sheet.
Feb 08	Read your lecture notes & do pgs 60-61: 2, 3, 11, 12. Read through the suggestions of study techniques from Jan 18 above and see which one(s) might work for you. Test 1 will be on Thursday; see Canvas for an information sheet.
Feb 10	Test 1 today; see Canvas for an information sheet. Read your lecture notes & 2^nd half of pg 59 & do pgs 60-61: 13(a), 16. Read pg 64 & pgs 908-909 & do pgs 65-66: 1, 5, 6, 14(a)(b), 17. Read Section 2.5 (pgs 66-71).
Feb 15	Read your lecture notes & pgs 75-85 & do pgs 85-89: 6, 7, 10. Look over the solutions to Test 1 on Canvas; use password "math5331" to open the file. (If you have trouble opening the pdf file, send an email to Dr. Vancliff about it.) Read through the suggestions of study techniques from Jan 18 above and see which one(s) might work for you.
Feb 17	In lecture today, we introduced the notion of left coset and right coset. By a normal subgroup of a group G, we mean a subgroup H of G that satisfies gH = Hg for all g ∊ G. Read your lecture notes & pgs 75-85 & do pgs 85-89: H 3, H 4, 22(a).
Feb 22	Read your lecture notes & pgs 75-85 & also watch the video re Section 3.1 in "After-Class Videos” in Canvas Modules. Do pgs 85-89: 4, 20, 3, 33, 36, 37, 40, 41.
Feb 24	LECTURE CANCELLED BY ORDER OF UTA (inclement weather). Lectures resume on Tuesday, on campus, in PKH 109.
Mar 01	Read your lecture notes & pgs 89-95 & also watch the video re Section 3.2-3.4 in "After-Class Videos” in Canvas Modules. Do pgs 95-96: 1, 6, 5, 4, 12 (hint: consider gh ↦ ? ), 16 (hint: see Corollary 9 in book). Read your lecture notes & pgs 97-100 & do pg 101: 8. Read pgs 102-105 & do pg 106: 1, 7 (hint: use the 4^th isomorphism theorem).
Mar 03	Read your lecture notes & pgs 106-111 & do H 5 & do pg 111: 2, 10. Test 2 will be 1 week from today; see Canvas for an information sheet. Look over Test 1 and its solutions on Canvas to study for Test 2; use password "math5331" to open the file. (If you have trouble opening the pdf file, send an email to Dr. Vancliff about it.)
Mar 08	Read lecture notes & pgs 223-228 & do pgs 230-233: 1, 11, 14(c), 28, 29, 30(a). Test 2 on Thursday; see Canvas for an information sheet. Look over Test 1 and its solutions on Canvas to study for Test 2; use password "math5331" to open the file. (If you have trouble opening the pdf file, send an email to Dr. Vancliff about it.)
Mar 10	Test 2 today; see Canvas for an information sheet. Recall that a subring of a ring R is a subgroup of (R, +) that is closed under the multiplication of R. Remark If S is a subset of a ring R, then, in order to show S is a subring of R, it suffices to show 1. S is nonempty, and 2. S is closed under subtraction, and 3. S is closed under multiplication. Note that 1 and 2 prove that S is a subgroup using the subgroup criterion. E.g., 2ℤ is a subring of ℤ. E.g., ℤ is a subring of ℚ. E.g., ℚ is a subring of ℝ. E.g., ℤ[i] = { a + bi ∊ ℂ : a, b ∊ ℤ, i² = –1} is a subring of ℂ. This subring is called the ring of Gaussian integers. E.g., {continuous functions f : ℝ → ℝ } is a subring of the ring {functions f : ℝ → ℝ }. In this example, the addition is given by (h + g)(x) = h(x) + g(x) and (hg)(x) = h(x)g(x) for all x ∊ ℝ and for all h, g ∊ {functions f : ℝ → ℝ }. E.g., {differentiable functions f : ℝ → ℝ } is a subring of the ring {functions f : ℝ → ℝ }. Addition and multiplication as in the previous example. E.g., {differentiable functions f : ℝ → ℝ } is a subring of the ring {continuous functions f : ℝ → ℝ }. Addition and multiplication as in the previous 2 examples. Read lecture notes & pgs 223-228 & do pgs 230-233: 5(c), 6(c), 7, 9. Watch the video re Section 7.2 in "After-Class Videos” in Canvas Modules & read pgs 233-236 & do pgs 237-239: 1, 3(b)(c), H 6. Optional: do pg 238: 4.
Mar 15 & Mar 17	SPRING BREAK Go over any tests and their solutions on Canvas & read over all lecture notes & get caught up on watching any lecture recordings/videos. Read through the suggestions of study techniques from Jan 18 above and see which one(s) might work for you. Get caught up on all homework!! Watch this fun TED talk on symmetry and algebra.
Mar 22	Read your lecture notes & pgs 233-237 & do pgs 237-239: 9-12. Look over the solutions to Test 2 on Canvas; use password "math5331" to open the file. (If you have trouble opening the pdf file, send an email to Dr. Vancliff about it.) Read through the suggestions of study techniques from Jan 18 above and see which one(s) might work for you.
Mar 24	Read your lecture notes & pgs 239-244 & pgs 246-247 & watch the video in "After-Class Videos” in Canvas Modules through the end of Section 7.3 and do pgs 247-251: 6, 7, 16, 32, 10(a)(b)(f), 18, 22, 34(a)(b) ( for the last question, note the definitions on pg 247 – in particular, IJ = { ∑ i_kj_k : where the sum is a finite sum & i_k ∈ I & j_k ∈ J ∀ k } ); also do H 7. Watch the rest of the video (which discusses some of Section 7.4) in "After-Class Videos” in Canvas Modules & read pg 251. If you found the TED talk linked above led to a broken link, retry it; it should (in theory) work now.
Mar 29	Read your lecture notes & pgs 251-256 & do H 8 & pgs 256-260: 8, 16(b), 4, 9 & H 9 & H 10, 6 (hint: be sure to show that any inverse is both a left inverse & a right inverse), 32(a), 37, 38 (hint: use #37), 5.
Mar 31	Read your lecture notes & pgs 260-264 (note the error on pg 263 in the Definition – see errata) & do pgs 264-265: 4 & H1 1 (note that the notation $R\setminus P$ means the elements of the ring R that are not in the ideal P), H1 2, H1 3 (optional) & H1 4 (optional). Test 3 will be 1 week from today; see Canvas for an information sheet. Look over Tests 1 & 2 and their solutions on Canvas to study for Test 3; use password "math5331" to open the solution files. (If you have trouble opening the pdf files, send an email to Dr. Vancliff about it.)
Apr 05	Read the remark on pg 279, the proof of Proposition 7 on page 280 and Corollary 8, and its proof, on page 281 and do pg 282: 3. Read your lecture notes & the proof of Proposition 10 on pg 284 and the proof of Proposition 12 on pg 286 and do pg 292: 1, 6(a). Read pgs 295-298 and do pgs 298-299: 1(a)(b), 2(a)(b), 6, 15, H1 5. Read Theorem 3 on pg 299 (Division Algorithm for F[x], where F is a field). (Some of you might prefer this proof of Corollary 4 on pg 300.) Do pg 301: 6(a)(b)(c) & pg 306: 4(d) & pgs 311-312: 7, 11. Hint: although some of these questions are in sections not covered in class, they can be solved using the material covered in class and the suggested readings. Test 3 on Thursday; see Canvas for an information sheet. Look over Tests 1 & 2 and their solutions on Canvas to study for Test 3; use password "math5331" to open the solution files. (If you have trouble opening the pdf files, send an email to Dr. Vancliff about it.)
Apr 07	Test 3 today; see Canvas for an information sheet. Watch the video in "After-Class Videos” in Canvas Modules on Section 10.1 (start of topic of modules) and read pgs 337, 339, 342-343. Do pg 343: 1, H16-H19 & pgs 343-4: 3, 5-7, 9, H 20 & pgs 344-345: 10, 20, 18. Optional: 13 & 22 on pgs 344-345. We will work on Sections 10.2 and 10.3 on Tuesday.
Apr 12	Read your lecture notes & pgs 345-349 and do pg 350: 1, 7, H2 1, H2 2 & read pgs 351-355 & do pg 356: 4, 8, 11 & H2 3 & H2 4. Look over the solutions to Test 3 on Canvas; use password "math5331" to open the file. (If you have trouble opening the pdf file, send an email to Dr. Vancliff about it.) Read through the suggestions of study techniques from Jan 18 above and see which one(s) might work for you.
Apr 14	Read your lecture notes & pgs 112, 114-115 & do pgs 116-117: 1, 6(a). Read pgs 118-120 & do pgs 121-122: 1(a), 14(hint: use Corollary 5), 8 (hint: do not assume \|G\| finite, but consider G acting on the left cosets of H in G).
Apr 19	Read your lecture notes & pgs 122-125. Do pgs 130-132: 25 (hint: uses Math 5333), 10(a)[hint: see Prop 10 on pg 125], 6, 13. Read pgs 133-135 and do pgs 137-138: 2, 7, 15. Read pgs 152-154 before April 26; this material will be assumed in the lecture on April 26.
Apr 21	Read your lecture notes & pgs 142-146 & watch the videos on Canvas under “Modules” called “Proof of Sylow’s Theorem”. Do pgs 146-147: 13-16, 18, 19, 30. Optional: 17 on pg 147. Read pgs 152-154 before April 26; this material will be assumed in the lecture on April 26.
Apr 26	Read your lecture notes & pgs 158-165 & do pgs 165-166: 1(a)-(c), 2(a)-(d), 3(a)-(d), 4(a),(b), H2 5, H26 & do pg 198: 8. Watch the video on Canvas under “Modules” called “Proof that Every Finitely Generated Abelian Group has a Presentation Matrix”.
Apr 28	Read your lecture notes & pgs 193-195 & Theorem 11 on pg 196 & do H2 7, H28. Remember to complete online the student feedback survey by 11 pm Tuesday May 3 -- check your mymav e-mail for the link. I appreciate the feedback (e.g., quality of website, quality of information sheets, quality of videos, quality of online lectures in 1^st 3 weeks of semester, use of ipad for lectures, use of 2 video feeds for online lectures, use of Canvas, tests’ solutions, quality of corny jokes,…..); thank you!
May 03	We will spend today's lecture reviewing the course material & discussing any questions from students. No new homework. Remember to complete online the student feedback survey by 11 pm Tuesday May 3 -- check your mymav e-mail for the link. I appreciate the feedback (e.g., quality of website, quality of information sheets, quality of videos, quality of online lectures in 1^st 3 weeks of semester, use of ipad for online lectures/videos, use of 2 video feeds for online lectures, use of Canvas, use of whiteboard in classroom, tests’ solutions, usefulness of homework, quality of corny jokes,…..); thank you! Regular office hour times do not apply after today. See announcements in Canvas for times of office hours after today.
May 05	FINAL TEST today, starting at 5:30 pm in PKH 109; see Canvas for an information sheet after April 21.

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