Jan 21

Attendance will be recorded, 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  https://vancliff.utasites.cloud/T/S09/1426study.pdf   and
https://www.jeffreybennett.com/pdf/How_to_Succeed_general.pdf for ideas on how to study most effectively. 
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).
Read lecture notes & pgs 16-21 & do pgs 21-23: 6(a)-(c), 8, 9, 12, 22, 25-27, 29, 30 (hint: consider division algorithm on pg 4 of book).
Do H1.

Jan 23

Read lecture notes & pgs 23-27 & do pg 28: 14, 16.
Read lecture notes & pgs 29-32 & do pgs 32-33: 1, 4(a).
Read lecture notes & pg 34 & do pg 35: 2, 3, 10, 11(a)(b).
Read pg 36 & do pg 36: 1.

Jan 28

Read lecture notes & pgs 36-39 & do pgs 39-41: 1, 13, 14, 15, 18, 20, 26.
Read lecture notes & pgs 41-44 & do pgs 44-45: 2, 16, 17.
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 30

Read lecture notes & pgs 41-44 & do pgs 44-45: 4, 6.
Read lecture notes & pgs 46-48 & do pgs 48-49: 1(a)(e), 2(d), 10, 11(a)(c), 15, & do   H2.

Feb 04

Read pgs 51-52 & do pgs 52-53: 2, 3, 10, 11, 12(a)-(c).
Read lecture notes & 2^nd half of pg 59 & do pgs 60-61: 2, 3, 11, 12.

Feb 06

Read 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).
Test 1 will be 1 week from today; see Canvas for an information sheet.

Feb 11

Read lecture notes & pgs 75-85 & do pgs 85-89: 6, 7, 10.
Test 1 will be on Thursday; see Feb 6 above for details.

Feb 13

Test 1 today; see Feb 6 above for details.
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 lecture notes & pgs 75-85 & do pgs 85-89: H3, H4, 22(a).

Feb 18

Read lecture notes & pgs 75-85 & do pgs 85-89: 4, 20, 3, 33, 36, 37, 40, 41.
Look over the solutions to Test 1 on Canvas in T1(score); 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 21 above and see which one(s) might work for you.

Feb 20

Read lecture notes & pgs 89-95 & do pgs 95-96: 1, 6, 5, 4, 12 (hint: consider gh ↦ ? ), 16 (hint: see Corollary 9 in book).

Feb 25

Read lecture notes & pgs 97-100 & do pg 101: 8.   Read pgs 102-105 & do pg 106: 1.

Feb 27

Read lecture notes & pgs 102-105 & do pg 106: 7.
Read pgs 106-111 & do   H5  & do pg 111: 2, 10.

Mar 03

Definition
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 ϵ ℂ : a, b ϵ 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 ϵ ℝ and for all h, g ϵ {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: 1, 11, 14(c), 28, 29, 30(a),, 5(c), 6(c), 7, 9.

Mar 05

Read lecture notes & pgs 233-237 & do pgs 237-239: 1, 3(b)(c), H6, 9-12. Optional: do pg 238: 4.

Test 2 will be 2 3 weeks from today (on Mar 19 on Mar 26); 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
&
Mar 12

SPRING BREAK
Go over Test 1 and its solutions on Canvas & read over all lecture notes from start of semester until now.
Read through the suggestions of study techniques from Jan 21 above and see which one(s) might work for you.
Watch this fun TED talk on symmetry and algebra.

Mar 17

Instruction moved online owing to covid-19 starting today. Check your Canvas notifications to check you can receive Canvas announcements and Canvas notification of conference initiation.

Read lecture notes, check Canvas for online videos and recording of lecture & read pgs 239-244 & pgs 246-247 &
do pgs 247-251: 6, 7, 16, 32, 10(a)(b)(f), 18.
Test 2 is on Thursday; see Mar 5 above for details. Moved to Mar 26 owing to instruction being moved online.

Mar 19

Read lecture notes, check Canvas for online videos and recording of lecture & read pgs 239-244 & pgs 246-247 &
do pgs 247-251: 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 H7.

Mar 24

Read lecture notes & pgs 251-256 & do H8 & pgs 256-260: 8, 16(b), 4,
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.

Test 2 is on Thursday; see Mar 5 above for details. Also, see the instructions emailed at the weekend about what to bring to Test 2 and how to practice in advance for Test 2 re camera and scanning/uploading solutions etc.

Mar 26

Test 2 today; see Mar 5 above for details.
Read lecture notes & pgs 251-256 & do pgs 256-260: 9, & H9 & H10.

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Mar 31

Look over the solutions to Test 2 on Canvas in T2(score); 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 21 above and see which one(s) might work for you.

Apr 02

Read lecture notes & pgs 260-264 (note the error on pg 263 in the Definition – see errata) & do pgs 264-265: 4  & H11, H12,
H13 (optional) & H14 (optional).
Read lecture notes, remark on pg 279 and Corollary 8, and its proof, on page 281.
Do pg 282: 3.

We are behind in the material and so a bit more reading is going to be assigned than usual…..
Read Section 8.3 through the end of page 286. Read the 1^st assertion of Theorem 14 on page 287 and the proof of that assertion.
Do pg 292: 1, 6(a).
Read pgs 295-298 and do pgs 298-299: 1(a)(b), 2(a)(b), 6, 15, H15.
Read Theorem 3 on pg 299 (Division Algorithm for F[x], where F is a field) and its proof and Corollary 4 on page 300 and the paragraph after Corollary 4. Some of you might prefer this proof of Corollary 4.
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.

On April 7, we will start the topic of modules.

Apr 07

Read lecture notes & pgs 337, 339, 342-343.  
Do pg 343: 1 & do H16-H19 & pgs 343-4: 3, 5-7, 9,   & H20 & pgs 344-345: 10, 20, 18.
Optional: 13 & 22 on pgs 344-345.

Apr 09

Read lecture notes & pgs 345-349 & watch the video on Canvas (under “Modules” & called “April 9 (required supplement)”).
(Remark: the meaning of the word “Module” used by Canvas does not equal the mathematical meaning of the word “Module”!!)
Do pg 350: 1, 7   & do H21, H22 & read pgs 351-355 & do pg 356: 4, 8, 11 & H23 & H24.

Apr 14

Read 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 16

Read 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 lecture notes and watch the video on Canvas (under “Modules” & called “April 16 (required supplement)”).
Do pgs 137-138: 2, 7, 15.

Test 3 will be 1 week from today; see Canvas for an information sheet (this test will have a different format than Test 2, so a practice test is provided in Canvas).
Look over Tests 1 & 2 and their solutions (see T1(score) & T2(score)) on Canvas to study for Test 3; use password "math5331" to open the files.
(If you have trouble opening a pdf file, send an email to Dr. Vancliff about it.)

Apr 21

Read lecture notes & pgs 142-146   &  watch the videos on Canvas under “Modules” & called “Proof of Sylow’s Theorem (required supplement)”;
those 2 videos are proofs of Sylow’s Theorem and last 30 minutes each.
Do pgs 146-147: 13-16, 18, 19, 30.   Optional: 17 on pg 147.
Read pgs 152-154 before April 28; this material will be assumed in the lecture on April 28.

Test 3 on Thursday/Friday; see April 16 for details.

Apr 23

Test 3 today/tomorrow; see April 16 for details.
No lecture today – you can choose to use this time to take Test 3.
Read pgs 152-154 before April 28; this material will be assumed in the lecture on April 28.

Apr 28

Read pgs 152-154, if not already done so.
Read lecture notes & pg 158.

Look over the solutions to Test 3 on Canvas in T3(score); use password "math5331" to open the file.
(If you have trouble opening the pdf file, send an email to Dr. Vancliff about it.)
Note that the files containing my handwritten comments on your solutions to Test 3 are stored in the comments section of T3(the test); in order to access those comments, please read the following. From the Canvas app, you should be able to access those comments from either the grades section of Canvas or from the Assignments section of Canvas. Once on the page T3(the test), click on "Submission and Rubric", and then click on “comments” in the bottom bar to see the files I uploaded or to see any typed comments (if there are any of the latter). From a browser (after logging into Canvas), you should be able to access the relevant files with my comments by going to “Grades” and then click on “T3(the test)” and that should take you to a page that provides a panel on the right side that provides links to the files and any typed comments (if there are any of the latter). Note that I have been told that accessing “T3(the test)” from Assignments in a browser does NOT provide that same side panel (instead it states your score etc but no access to the files).

Read through the suggestions of study techniques from Jan 21 above and see which one(s) might work for you.

Apr 30

Read lecture notes & pgs 158-165  & do pgs 165-166: 1(a)-(c), 2(a)-(d), 3(a)-(d), 4(a),(b),   H25, H26   &   do pg 198: 8.

May 05

Watch the video on Canvas (under “Modules” & called “May 5 (required supplement)”).
Read lecture notes & pgs 193-195 & Theorem 11 on pg 196   &   do   H27, H28.
Complete online the student feedback survey by 11 pm on Fri May 8 -- check your mymav e-mail for the link.
I appreciate the feedback (e.g., quality of website? examples provided in lecture? use of whiteboard in classroom? tests’ solutions? usefulness of homework? interactive online lectures?; corniness of jokes?!); thank you!

May 07

We will spend today's lecture finishing up the material and discussing any questions from students; no new homework.
This is the last lecture of Math 5331.
Complete online the student feedback survey by 11 pm on Fri May 8 -- check your mymav e-mail for the link.
I appreciate the feedback (e.g., quality of website? examples provided in lecture? use of whiteboard in classroom? tests’ solutions? usefulness of homework? interactive online lectures?; corniness of jokes?!); thank you!

Regular office hour times do not apply next week. See the calendar in Canvas for office hours next week.

May
13-14

FINAL TEST: official time window = 5:30-8:00 pm on May 14. However, since the test will be online (owing to COVID-19), it will be accessible starting Wednesday May 13 at 8:30 pm until Thursday May 14 at 8:30 pm; in particular, a student can choose to take the test during the official time window if s/he wishes. See Canvas for an information sheet.