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/study.pdf   and
read    https://www.jeffreybennett.com/pdf/How_to_Succeed_general.pdf  for ideas on how to study most effectively.

Read lecture notes & do Sec 18 pgs 174/5: 1, 3, 5, 7, 9, 11.

Jan 23

Read lecture notes, pages 167-174 & do Sec 18 pgs 175/6: 31, 38,
46 (hint: think binomial theorem from Math 1302; or see here), 22, 27, 23.  (order of questions is not random; recall remark above)

Jan 28

Read lecture notes, pages 167-174 & do Sec 18 pgs 175/6: 14, 16, 19, 20, 40.

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 & Sec 19 & do Sec 19 pgs 182/3: 14, 23, 24, 28, 5-12, 27, 29.
The Quiz will be 1 week from today; see Canvas for an information sheet.

Feb 04

Read lecture notes & Sec 20 up to pg 187.5 & do Sec 20 pgs 189/90: 1-3, 24, 5, 23a-h (justify answers).
The Quiz will be on Thursday; see Jan 30 above for details.

Feb 06

Quiz today; see Jan 30 above for details.
Read lecture notes & Sec 21 & do Sec 21 pgs 196-7: 1, 2, 7, 8,, 12, 14.
Test 1 will be 1 week from today; 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.
(If you have trouble opening the pdf file, send an email to Dr. Vancliff about it.)

Feb 11

To finish Section 22, notice that the kernel of the evaluation map Φ_α consists of those polynomials that evaluate to zero on α.
So if, e.g., the field F = ℚ and the field E = ℝ, then x² + x - 6 belongs to the kernel of Φ₂, since x² + x - 6 evaluated
at α = 2 is zero. Moreover, also notice that it is possible for a polynomial f ϵ F[x] to have no zero at all in the field F;
e.g., f = x² - 2 ϵ ℚ[x] has no zero in ℚ. Whatever the field, we often write f(α) for Φ_α(f).
This finishes Section 22.
Read lecture notes & Sec 22 & do Sec 22 pgs 207-8: 1-6, 22, 25,, 7, 8, 14, 20.
Test 1 will be on Thursday; see Feb 6 above for details.

Feb 13

Test 1 today; see Feb 6 above for details.
Read lecture notes & pgs 209-213 & do Sec 23 pgs 218-220: 2, 4, 35, 36, 9.

Feb 18

Read lecture notes & pgs 209-213 & do Sec 23 pgs 218-220: 10, 6, 34, 14, 16.
Look over the solutions to Test 1 on Canvas in T1(score); use password "math4321" 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 209-213 & do Sec 23 pgs 218-220: 18, 27, 28.
Read lecture notes & pgs 224.5-226 & do Sec 24 pgs 226-7: 14, 5, 4, 6, 10.

Feb 25

Read lecture notes & Sec 26 & do Sec 26 pgs 243-5: 2, 17, 18, 29, 37, 38(a)(b), 4.

Feb 27

Read lecture notes & Sec 26 & do Sec 26 pgs 243-5: 12-14, 24 (hint: see #18),, 22(a)(c), 25, 26, 32.
Read lecture notes & do Sec 27 pgs 254: 34, 35, 37.

Mar 03

Theorem
Let R be a nonzero commutative ring with unity. If I is a proper ideal of R, then I is a prime ideal of R if and only if R/I is an integral domain.
Proof
(⇐) Suppose ab ϵ I, where a, b ϵ R. So (a+I)(b+I) = ab + I = 0+I, where the last equality is a consequence of ab ϵ I.
Since R/I is an integral domain, it follows that a+I = 0+I or b+I = 0+I; in other words, a ϵ I or b ϵ I. Since a and b were arbitrary, it follows that I is a prime ideal.
(⇒) Suppose xy = 0 in R/I, where x, y ϵ R/I. There exist a, b ϵ R such that x = a+I and y = b+I. Substituting, we have
(a+I)(b+I) = xy = 0 , where 0 is the zero coset, so is 0+I. Thus, we have that ab + I = 0 + I, which implies that ab ϵ I.
Since I is a prime ideal of R, it follows that a ϵ I or b ϵ I. This implies that a+I = 0+I or b+I = 0+I, so x = 0 or y = 0.
Since x and y were arbitrary in R/I, we can conclude that R/I is an integral domain.

E.g., ℤ/{0} ≅ ℤ, which is an integral domain. So, it follows from the last theorem that {0} is a prime ideal in ℤ.
(However, {0} is not a maximal ideal of ℤ.)

Corollary
Let R be a nonzero commutative ring with unity. If M is a maximal ideal of R, then M is a prime ideal of R.
Proof
Since M is maximal, R/M is a field and so R/M is an integral domain, which implies that M is a prime ideal.

Read lecture notes & pgs 245-7 & learn box on pg 248.
Do Sec 27 pgs 252-4: 1 (max ideals only), 2 (max ideals only), 1, 2, 24.

Mar 05

Read this file which is a summary of some of today’s material but written for Spring 2019, so we might have covered some of the proofs in class, contrary to what it says in that file. Read lecture notes & pgs 245-7 & learn box on pg 248.
Do Sec 27 pgs 252-4: 14(a)(b)(f), 31, 5, 6, 30.

Test 2 will be 2 3 weeks from today (on Mar 19 on Mar 26); see Canvas for an information sheet.
Look over the Quiz & Test 1 and their solutions on Canvas to study for Test 2; use password "math4321" to open the files.
(If you have trouble opening the pdf files, send an email to Dr. Vancliff about it.)

Mar 10
&
Mar 12

SPRING BREAK
Go over the Quiz & Test 1 and their solutions on Canvas & read over all lecture notes from start of semester until now.
(If you have trouble opening the pdf files, 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.
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 read pgs 265-9 & do
Sec 29 pgs 272-4: 1, 2, 4, 11, 12, 14, 6 (Hint: E. Criterion), 21, 27.
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 and read this file (which was discussed in class) & pgs 265-9 & do Sec 29 pgs 272-4: 13, 15, 18, 30, 31.

Mar 24

Recall: if F ≤ E are fields, and if α ∊ E is algebraic over F and n = deg (α, F) , then our work in class today
implies that {1, α , α², …., α^n-1 } is a basis for F(α) over F.
The last bit of Section 30 that we did not have time to cover in “class” is now on Canvas as a video; see March 24 under Modules.
Read lecture notes & Sec 30 & do Sec 30 pgs 280-2: 15(a)-(d) (justify), 25 (note: "transformation" = "map"), 1, 2, 4, 6, 7, 19.
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 283 & 284, especially the Tower Theorem, & do
Sec 31 pgs 291-3: 1, 7 & do degree only for 2, 3, 5, 9, 11, 13.

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Mar 31

Read lecture notes & pgs 283-289 & do Sec 31 pgs 291-3: 2, 3, 5, 9, 11, 13, 19(a)-(c), 22, 28, 29.

Look over the solutions to Test 2 on Canvas in T2(score); use password "math4321" 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.

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 02

Read lecture notes & pgs 283-289 & do
Sec 31 pgs 291-3: 35 (hint: show that the squaring function is neither one-to-one nor onto).
Optional exercise: compare Zorn's Lemma on page 289 with the ``version’’ of Zorn's Lemma in the episode of
The Simpsons that aired on Sunday April 5, 2015; are there any mistakes in the ``version’’ in that episode?
Read lecture notes and pg 300 and do Sec 33 pgs 304-5: 8(g)(justify), 1.

Apr 07

Read lecture notes, this file & Sec 33 & do Sec 33 pgs 304-5: 8(a)(justify), 11, 8(c)(justify), 4. (Hint: for #4:
there are many ways to do this question. You could read the statement of Thm 33.10 and either use Euler's φ function (why?),
or 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 29 #18).)

Apr 09

Read lecture notes, page 1 of this file & watch the video on Canvas (under “Modules” & called “April 9: Sec 48”) & read Sec 48
& do Sec 48 pgs 422-4: 1-3, 7-11, 13, 25,, 26, 15-21, 36(a),(b).

Test 3 will be 1 week from today; see Canvas for an information sheet.
Look over the quiz, Test 1 and Test 2 and their solutions (see T1(score) and T2(score)) on Canvas to study for Test 3; use password "math4321" to open the files.
(If you have trouble opening a pdf file, send an email to Dr. Vancliff about it.)

Apr 14

Read lecture notes, this file & Sec 49 & do Sec 49 pgs 430-1: 1-3, 7, 8(b)(d)(e) (justify), 13.
Test 3 on Thursday/Friday; see April 9 above for details.

Apr 16

Test 3 today/tomorrow; see April 9 above for details.
No lecture today – you can choose to use this time to take Test 3.

Apr 21

Read lecture notes & Sec 50 & do Sec 50 pgs 434-6: 1-5, 14(a)-(e),(j) (justify), 18, 21.

Look over the solutions to Test 3 on Canvas in T3(score); use password "math4321" 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 23

Read lecture notes & Sec 51 & do Sec 51 pgs 442-4: 7, 15, 1-4, 9 ( hint: use E = F(α, β) ), 13,, 8(a)-(g)(justify).

Apr 28

Read lecture notes & Sec 53 & do Sec 53 pgs 454-6: 1, 2, 11(a)(b), 13.

Apr 30

Look over today's handout; this handout can also be found in Canvas under Modules and is called “ April 30 "handout" “.
Read lecture notes & Sec 53 & do Sec 53 pgs 454-6: 3, 4, 8, 11, 15(a)-(c),(e),(g) (justify).
Read Sec 56.

May 05

We will spend today's lecture reviewing the course material & discussing any questions from students; no new homework.
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 discussing any questions from students; no new homework.
This is the last lecture of Math 4321.
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
11-12

FINAL TEST: official time window = 2:00-4:30 pm on May 12. However, since the test will be online (owing to COVID-19), it will be accessible starting Monday May 11 at 7 pm until Tuesday May 12 at 7 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.