Homework for MATH 4321 in Spring 2025 – Website of Dr. Michaela Vancliff

return to Spring 2025 MATH 4321 webpage

Recommended textbook
(required):

A First Course in Abstract Algebra, 8^th Ed., by J. B. Fraleigh & N. E. Brand,
Pearson, 2021.

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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 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!
The homework from Spring 2023 can be viewed here; in particular, it will give you a rough outline of what material will be presented and how many lectures will be spent on each item.
Recall that our test dates are: Feb 1-5 (online), Feb 6, Mar 6, April 10 and May 6.
LAST REVISION: 4/08/25.

The homework from Fall 2023 can be viewed in its entirety here.

Jan 14	Check your Canvas notifications to check you can receive Canvas announcements. 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. Review the course’s Canvas portal. Read this study tip and read https://www.jeffreybennett.com/pdf/How_to_Succeed_general.pdf for ideas on how to study most effectively.
Jan 16	Read your lecture notes (meaning the notes you should have taken during lecture) and Section 22, & do Sec 22 pgs 191-3: 1, 3, 5, 7, 9, 11, 33, 40, 48 (hint: think binomial theorem from Math 1302; or see here), 27 (order of questions is not random; recall remark above).
Jan 21	Read your lecture notes and Section 22. We did not quite finish Section 22, but the part we did not cover is as follows. Definition A subring of a ring R is a subset S of R that is itself a ring using the same operations of + and multiplication used in R. We write S ≤ R. A subfield of a field F is a subset S of F that is itself a field using the same operations of + and multiplication used in F. We write S ≤ F. The proof of claim 1 in our proof that the special ring D in lecture is a division ring shows us how to prove that a subset S of a ring R is a subring; namely, one only needs to prove that: (a) 0 ∈ S, and (b) a − b ∈ S for all a, b ∈ S, and (c) ab ∈ S for all a, b ∈ S, in order to prove that S is a subring of R. Example 2ℤ ≤ ℤ (a), (b) and (c) hold for 2ℤ. Nonexample S = {all odd integers} is not a subring of ℤ since 0 ∉ S (alternatively: a = 5, b = 3, a − b = 5 − 3 ∉ S ). Do Sec 22 pgs 192-3: 22, 23, 14, 16, 19, 20, 42, 39.
Jan 23	Read your lecture notes & Sec 23 & do Sec 23 pgs 198-200: 20, 29, 30, 34, 11, 14-16. The Quiz will open in 1 week; see Canvas for an information sheet.
Jan 28	Read your lecture notes & Sec 23 & do Sec 23 pgs 198-200: 5-10, 17-18, 33, 35. Watch the video “Section 24” in Canvas Modules and read Sec 24 through pg 202 & do Sec 24 pgs 204-5: 1-3, 24, 5, 23a-h (justify answers). Optional: I also suggest that y’all skim Section 25 to see how Fermat’s Little Theorem is used in cryptology. If we have time at the end of the semester, we will briefly discuss Section 25. The Quiz will open on Friday; see Canvas for an information sheet.
Jan 30	Read your lecture notes & Sec 26 & do Sec 26 pgs 216-217: 1, 2, 7, 8,, 12, 14. The Quiz opens tomorrow; see Canvas for an information sheet. Read through the suggestions of study techniques from Jan 14 above and see which one(s) might work for you. 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. Test 1 will be 1 week from today; see Canvas for an information sheet.
Feb 04	Read your lecture notes & Sec 27 & Sec 28 & do Sec 27 pgs 225-227: 1-6, 22, 25,, 7, 8, 14, 20 and Sec 28 pgs 235-237: 2, 4, 35, 36. The Quiz is due tomorrow; see Canvas for an information sheet. Look over the quiz and its solutions on Canvas to study for Test 1; use password “math4321” to open the file. Read through the suggestions of study techniques from Jan 14 above and see which one(s) might work for you. Test 1 will be on Thursday; see Canvas for an information sheet.
Feb 06	Test 1 today; see Canvas for an information sheet. Read your lecture notes & Sec 28 & do Sec 28 pgs 235-237: 9, 10.
Feb 11	Read your lecture notes & Sec 28 & do Sec 28 pgs 235-237: 6, 34 (hint: try small values of p first), 14, 27, 28. Look over the solutions to Test 1 on Canvas; use password “math4321” to open the file. Read through the suggestions of study techniques from Jan 14 above and see which one(s) might work for you.
Feb 13	Read your lecture notes & the last half of pg 25 from the instructor’s lecture notes (see “Relevant Links” in the Modules section of our Canvas portal) and read Sec 28 & do Sec 28 pgs 235-237: 16, 18.
Feb 18	Read your lecture notes & do Sec 32 pg 264: 14, 5, 4, 6, 10.
Feb 20	Read your lecture notes & Sec 30 & do Sec 30 pgs 248-250: 17, 37, 29, 18,, 26, 31, 30, 34,, 4, 12-14, 24 (hint: see #18), 25, 32.
Feb 25	Read your lecture notes & pgs 250-252 & do Sec 31 pgs 256-8: 25, 40, 36, 37, 39.
Feb 27	Read your lecture notes & Section 31 & learn the 3 facts on page 253 (just prior to Prime Fields). Do Sec 31 pgs 256-8: 1, 2, 26, 14(a)(b)(f), 15-19, 33, 5, 6, 32. Test 2 will be 1 week from today; see Canvas for an information sheet. Read through the suggestions of study techniques from Jan 14 above and see which one(s) might work for you.
Mar 04	Read your lecture notes & Section 31 & Section 33. Test 2 will be on Thursday; see Canvas for an information sheet. Read through the suggestions of study techniques from Jan 14 above and see which one(s) might work for you.
Mar 06	Test 2 today; see Canvas for an information sheet. Read your lecture notes & Section 33, and watch the video “Section 33” in Canvas Modules. Do Sec 33 pgs 273-5: 15(a)-(d) (justify), 25 (note: “linear transformation” is defined in #24), 1-6, 9, 19.
Mar 11 & Mar 13	SPRING BREAK Go over the Quiz and Test 1 and their solutions on Canvas & read over all lecture notes & get caught up on watching any lecture recordings/videos. Also watch this fun TED talk on symmetry and algebra (approx. 18 minutes). Read through the suggestions of study techniques from Jan 14 above and see which one(s) might work for you. Get caught up on all homework!!
Mar 18	Look over the solutions to Test 2 on Canvas; use password “math4321” to open the file. Read through the suggestions of study techniques from Jan 14 above and see which one(s) might work for you.
Mar 20	Read your lecture notes & Sec 39 and do Sec 39, pgs 317-319: 1, 2, 4, 11, 12, 14, 6 (hint: E. Criterion), 21, 27, 13, 15.
Mar 25	Read your lecture notes & Sec 39 and do Sec 39, pgs 317-319: 18, 30, 31.
Mar 27	Read your lecture notes & Sec 40, especially the Tower Theorem. Do Sec 40 pgs 327-8: 1, 7, 2, 3, 5, 9, 11, 13, 19(a)-(c), 22, 28, 29, 35 (hint: show that the squaring function is neither one-to-one nor onto). Optional exercise: compare Zorn’s Lemma on page 325 with the “version’’ of Zorn’s Lemma in the episode of The Simpsons that aired on Sunday April 5, 2015 (S26,E11); are there any mistakes in the “version’’ in that episode? Warning: it is very important that you keep up with the homework at this point; there will be a lot of new concepts and you will need fluency with the terminology and notation.
Apr 01	Read your lecture notes & Sec 42 and watch the video “A Few Results from Section 42” in Canvas Modules. Do Sec 42 pgs 339-340: 8(g)(justify), 1, 9 (hint: see pg 336), 8(a)(justify), 11, 8(c)(justify), 4 (hint: there are many ways to do this question: (i) you could read the statement of Thm 42.10 and either use Euler’s φ function (why?), or (ii) you could write down a generic field with 9 elements using an element α for the generator of the cyclic group of nonzero elements in the field and work with the nonzero elements of the field as powers of α (we did something like this 2nd method in class with the example drawn from Sec 39 #18)), 15(a)-(f).
Apr 03	Read your lecture notes & Sec 43. The Conjugation Isomorphism Theorem Let ${\color{White}E}$ be an extension field of a field ${\color{White}F}$ . If ${\color{White}\alpha,\beta\in E}$ are algebraic over ${\color{White}F}$ where ${\color{White}\deg(\alpha,F)=n,}$ then the map ${\color{White}\psi_{\alpha,\,\beta}:F(\alpha)\to F(\beta)}$ defined by ${\color{White}\psi_{\alpha,\,\beta}(\sum_{i=0}^{n-1}c_i\alpha^i)=\sum_{i=0}^{n-1}c_i\beta^i,}$ where ${\color{White}c_i\in F}$ for all ${\color{White}i,}$ is an isomorphism from ${\color{White}F(\alpha)\twoheadrightarrow F(\beta)}$ if and only if ${\color{White}\alpha}$ and ${\color{White}\beta}$ are conjugate over ${\color{White}F}$ . The proof of this result and two examples are provided in the online lecture notes. Theorem Let ${\color{White}E}$ be an extension field of a field ${\color{White}F}$ , and suppose that ${\color{White}\alpha\in E}$ is algebraic over ${\color{White}F}$ . (a) Every isomorphism ${\color{White}\psi:F(\alpha)\twoheadrightarrow}$ subfield of ${\color{White}E}$ such that ${\color{White}\psi\|_F=}$ identity on ${\color{White}F}$ maps ${\color{White}\alpha}$ onto a conjugate element ${\color{White}\beta}$ of ${\color{White}\alpha}$ over ${\color{White}F}$ . (b) For each conjugate element ${\color{White}\beta}$ of ${\color{White}\alpha}$ over ${\color{White}F}$ , there exists a unique isomorphism ${\color{White}:F(\alpha)\twoheadrightarrow\;}$ subfield of ${\color{White}E}$ that fixes ${\color{White}F}$ and maps ${\color{White}\alpha}$ onto ${\color{White}\beta}$ (and the previous theorem implies that this unique isomorphism is ${\color{White}\psi _{\alpha,\,\beta}}$ ). The proof of this result is provided in the online lecture notes and is proved in the video “One Result from Section 43” provided in the Modules section of Canvas. Do Sec 43 pgs 347-349: 1-3, 7, 8, 9-13, 25, 16-21, 38(a)(b). Test 3 will be 1 week from today; see Canvas for an information sheet. Read through the suggestions of study techniques from Jan 14 above and see which one(s) might work for you.
Apr 08	Read your lecture notes and do Sec 44 pgs 355-357: 1-6, 10, 20(c)-(f),(j) (justify), 24, 26. Test 3 will be on Thursday; see Canvas for an information sheet. Read through the suggestions of study techniques from Jan 14 above and see which one(s) might work for you.
Apr 10	Test 3 today; see Canvas for an information sheet.
Apr 15	Note that if ψ : F ⟶ F′ is a field homomorphism, and if φ : E ⟶ E′ is another field homomorphism, where F ≤ E and F′ ≤ E′, such that φ\|_F= ψ, we say that ψ extends to E and that φ is an extension of ψ to E. Read your lecture notes & Sec 44 and watch the 3 videos on Sec 44 in Canvas Modules. Do Sec 44 pgs 355-357: 11-13, 17, 22 (see the last 2 videos), 30 (see the last video). Look over the solutions to Test 3 on Canvas; use password “math4321” to open the file. Read through the suggestions of study techniques from Jan 14 above and see which one(s) might work for you.
Apr 17	Read your lecture notes & Sec 45 and do Sec 45 pgs 363-364: 7, 8(a)-(c),(i) (justify), 9, 12, 13, 1-4.
Apr 22	???
Apr 24	Look over today’s handout; this handout can also be found in Canvas under Modules > Relevant Links, and is called “April 24 handout”. Read your lecture notes, Sec 46 & Sec 49 and do Sec 46 pgs 369-371: 1, 2, 11(a)-(c), 13,, 3, 4, 8, 11, 15(a)-(c),(e),(g) (justify). Please remember to complete online the student feedback survey by 11 pm on Apr 29 — check your mymav e-mail for the link. I appreciate the feedback (e.g., use of website, the use of Canvas, the choice of homework questions from the book and/or any questions not from the book, the information sheets and/or the solution sheets for the tests, the examples provided in class, use of any videos, jokes sufficiently corny??….??). Thank you!
Apr 29	We will spend today’s lecture reviewing the course material & discussing any questions from students. No new homework. Please remember to complete online the student feedback survey by 11 pm today — check your mymav e-mail for the link. I appreciate the feedback (e.g., use of website, the use of Canvas, the choice of homework questions from the book and/or any questions not from the book, the information sheets and/or the solution sheets for the tests, the examples provided in class, use of any videos, jokes sufficiently corny??….??). Thank you! The Final Test will be on Tuesday May 6; see Canvas for an information sheet. Read through the suggestions of study techniques from Jan 14 above and see which one(s) might work for you. I will have my usual office hours through today inclusive; after today, I will have office hours as noted below.
Apr 30	Office hour today 4 pm – 6 pm in PKH 462. This is time the instructor is planning to be in PKH 462 for students to drop by to ask questions.
May 01	Office hour today 2 pm – 4 pm in PKH 462. This is time the instructor is planning to be in PKH 462 for students to drop by to ask questions.
May 05	Office hour today 1 pm – 2 pm online only – see Canvas announcements for the Teams link (same link as used for Luke’s office hours). This is time the instructor is planning to be on Teams for students to “drop by” to ask questions.
May 6 2:00 PM	FINAL TEST today, starting at 2 PM; see Canvas for an information sheet. Look over Tests 1-3 and the Quiz and their solutions posted on Canvas when you study for this test. (Pay attention to how the solutions on the solution sheets are written; e.g., the level of detail provided in each solution.) Read through the suggestions of study techniques from Jan 14 above and see which one(s) might work for you.