## Basic Course Information / Links

Course Sections (click on the section number for syllabus)

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

eLC

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 - 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).
• 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.
• 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].
• 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.
• 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.

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 25TEST 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!

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