## Basic Course Information / Links

**Course Sections (click on the section number for syllabus)**

Section 52037 / 52038: MTWRF 9:15am-10:15am in Boyd 323

**Useful Website Links**

Tutoring resources (includes some paid tutor options)

- The math department offers some free study halls many afternoons in Boyd, though I don't know how useful they are for 4000-level courses.
- (I'm not sure how tutor availability differs for the summer as well.)

**Class Schedule**

All textbook sections come from the course textbook. Your homework sets also use many problems from that textbook.

- The problem text may be recopied in homework assignments, so that you can do problems without having to keep your book open all the time.
- For more details, consult the syllabus.

Don't forget about handouts that are attached in this calendar, such as group-work solutions and HW assignments.

- Partial solution sets can instead be found on eLC.

Expect this calendar to be updated several times a week.

**UNIT 1: Fundamentals of Z, gcds, and modular congruence**

*Week 1:*

*Tues Jun 5, 2018:*Section 1.1. Introduction to basic + and * laws, recap of proof by induction.*Wed Jun 6:*Section 1.1. Induction practice, the Binomial Theorem.*Thu Jun 7:*Section 1.2. Divisors, Euclidean division (quotient and remainder), common divisors.*Fri Jun 8:*Section 1.2. Euclidean Algorithm practice, linear combinations ax + by for integers x,y.

**HW 1 - due Monday Jun 11**

- Every student must hand in all the required problems. Students in MATH 6000 must do at least one challenge problem.
- Any challenge problems completed beyond this minimum are counted as extra credit.

If you're having a hard time getting the textbook, here's the first section's homework to tide you over for the moment.

*Week 2:*

*Mon Jun 11.*Section 1.2. Relatively prime numbers gcd(a,b) = 1, primes, the Fundamental Theorem of Arithmetic.*Tues Jun 12.*Section 1.3. Congruence mod m, basic simplifications, some squares mod m.*Wed Jun 13**.*Section 1.3. Using mods to establish contradictions, Z_m, solving linear congruences ax = b (mod m)*Thu Jun 14.*Sections 1.3 & 1.4. Chinese Remainder Theorem, the definition of a ring.*Fri Jun 15.*Section 1.4. Some basic ring properties of 0 and 1, units (invertible elements) and zero divisors.**HW 2 is due today.**- Group-work

**HW 2 - due Friday Jun 15**

**TEST 1 will be Wednesday June 20.**

- Expect 5 to 7 questions, similar to HW level of difficulty, for a 60-minute test.
- The first question always asks you to repeat definitions or to give very short examples.

- This test covers material from Sections 1.1 to 1.4.
- Section 2.1 won't show up on the test, but Section 1.4 will (despite not appearing on HW 1 or 2).
- Make sure to go over your graded comments on HW sets!

- Calculators will be allowed to help you do basic divisions, but you still have to show your steps for approaches like the Euclidean Algorithm.
- Here's a topic list for the test.
- Here are practice problems for the test.
- We'll have an in-class review day, to go over your open questions or the practice problems, on Tuesday June 19.

*Week 3:*

*Mon Jun 18:*Section 2.1. Ordered rings.*Tues Jun 19:*Test 1 Review- Review materials, such as practice problems, are located in the previous week's links.

*Wed Jun 20:***TEST 1***Thu Jun 21:*Section 2.3. Basic computation in C, conjugates and magnitudes, polar form.- HW 3 is released now, due on Tues Jun 26.

*Fri Jun 22:*Section 2.3. Using polar form and DeMoivre's Theorem to determine roots of complex numbers.

**HW 3 - due Tues Jun 26** (starts after Test 1)

**UNIT 2: Complex numbers, polynomials, irreducibility tests**

*Week 4:*

*Mon Jun 25.*Section 2.4. Quadratic and cubic equations in C.*Tue Jun 26.*Section 3.1. Introduction to polynomials R[x], units in F[x], long division of polynomials.**HW 3 is due today.**- Group-work

- Wed Jun 27. Section 3.1. The Root-Factor Theorem, GCD and the Euclidean Algorithm in F[x].
*Thu Jun 28*. Section 3.1. Irreducible elements of F[x], Fundamental Theorem for F[x], a result about partial fraction decompositions.*Fri Jun 29*. Section 3.3 (day 1). Checking polynomials in Q[x] for rational roots, factoring via the Method of Undetermined Coefficients.**HW 4 is due today.**Start HW 5 as well!- You may want to read ahead in Section 3.3 to help with HW 5! Eisenstein's Criterion is especially useful.

**HW 4 - due Fri Jun 29**

**HW 5 (shorter) - due Tues Jul 3**

- You may want to read ahead in Section 3.3 in the book (especially Eisenstein's Criterion) over the weekend.

*Week 5:*

*Mon Jul 2.*Section 3.3. Using mods and primes to study irreducibility: Eisenstein's Criterion.*Tue Jul 3.*Sections 3.1 and briefly introduce 3.2. Calculations with polynomial mods, set the stage for F[alpha].**HW 5 is due today.**(See its download link above.)- Group-work

**No class on Wed Jul 4 (national holiday).***Thu Jul 5.*Test 2 Review. (See links below.)*Fri Jul 6.***TEST 2**

**TEST 2 is on Friday, July 6.**

- Like Test 1, expect 5 to 7 questions in 60 minutes, with definitions and/or examples coming first.
- This can include restating a theorem precisely. (See the practice test.)

- This test covers Sections 2.3, 3.1, and 3.3.
- While it is nice to know the quadratic formula, other than that, Section 2.4 is not covered.

- Calculators are allowed again.
- Here's a topic list for the test.
- Here are some practice questions for this test.
- We will go over review on Thursday July 5.

**UNIT 3: Vector spaces, adjoining elements to create F[alpha], ring ideals and the Fundamental Homomorphism Theorem**

*Week 6:*

*Mon Jul 9*. Section 5.1 (and brief recap of material from Tue Jul 3). Vector spaces over a field F, subspaces, span of a set of vectors.*Tues Jul 10*. Section 5.1. Linear independence, basis, and dimension.*Wed Jul 11*. Section 5.1 and Section 3.2. Linear maps, the definition of F[alpha], field degree [K : F] = dim(K) over F, "Degree Extension Formula" (Theorem 1.3).*Thu Jul 12*. Section 3.2. Proving F[alpha] is a field, computations in F[alpha] using minimal polynomials.- HW 6 is due today.

*Fri Jul 13*. Section 3.2. More practice with F[alpha]. Comparing different extensions F[alpha] and F[beta].

**HW 6 - due Thu Jul 12**

**HW 7 - due Tue Jul 17**

- This homework deals with pretty difficult ideas: I suggest office hours before the weekend to clarify key ideas!
- The handout for Friday's class should be really useful here, though it's a lot to process. (The book doesn't present these ideas as systematically as I'd like.)

*Week 7:*

*Mon Jul 16*. Section 3.2, and start 4.1. More practice with F[alpha,beta], splitting fields. Definition of morphism.*Tue Jul 17*. Section 4.1. Morphisms examples involving F[alpha], definition of ideal I <= R.**HW 7 is due today.**- Group-work

*Wed Jul 18*. Section 4.1. Principal ideals <a>, maximal ideals, congruence mod I.*Thu Jul 19*. Sections 4.1 and 4.2. The quotient ring R/I, definition of isomorphism.*Fri Jul 20*. Section 4.2. More isomorphism examples, Fundamental Homomorphism Theorem (FHT).

**HW 8 - due Mon Jul 23**

- This is the last HW set of the course. You get a while to work on it, but it has a lot of abstract ideas!
- Make sure to look over handouts above, including group-work solutions. They should help a lot.

**TEST 3 is on Wednesday, July 25.**

- As with Test 2, expect 5 to 7 questions, with at least one question on definitions or examples.
- This test covers Sections 3.2, 4.1, 4.2, and 5.1.
**You will be asked to repeat the proof of a theorem from class!**Look at the practice questions document below for more detail.

- Calculators are allowed.
- Here's a topic list for the test.
- Here are some review problems for this test.

*Week 8:*

*Mon Jul 23*. Miscellaneous topics from Sections 5.3 and 4.2. Finite fields of size p^n, examples of product rings R x S.- This is not covered on Test 3, but some examples with R x S could be on the final.
- (Also, the skills used in this day's notes should be good review / enrichment in preparation for Test 3.)

*Tue Jul 24*. Test 3 Review (see links above in Week 7).*Wed Jul 25*.**TEST 3***Thu Jul 26*. Section 5.2. Constructible numbers with straightedge and compass.- If this shows up on the final exam, it will only be one short question.

*Fri Jul 27*. Review day for the final exam!- Here are a few more practice problems, with solutions. (This won't cover all the exam topics though.)
- Expect 11 to 13 questions of similar difficulty to previous exams.

Details about the final exam are below.

Final exam details

Mon July 30, 2018

8:00am to 11:00am (try to show up a bit early to get yourself settled)

Boyd 323 (our usual classroom)

- This exam will cover all the material from the three in-class exams.
- Expect somewhere on the order of 11 to 13 questions, fairly similar to previous test questions. This should be around twice the length of an in-class test.

- No bathroom breaks once the test starts. Talk to me if this is an issue.
- You'll have to leave your backpack (or similar containers) at the side of the room during the test. Turn phones and smart watches off.

Last updated: 7/27/2018