Class Schedule

 

All textbook sections come from the official course textbook, Abstract Algebra: A Geometric Approach by Theodore Shifrin. Your homework sets also use many problems from that textbook. For more details, consult the syllabus.

 

Expect this calendar to update weekly. At the end of each week, we will usually have an extra handout which can either show you more examples or can go through a detailed example more thoroughly. Make sure to check this calendar at least on Mondays and Fridays.


Week 1

Day

Book Sections

Topics / Due Dates

Monday Jan 5

1.1

Intro to course, algebra law demonstration, quick recap of induction

Wednesday Jan 7

1.1

Induction practice, strong induction, well-ordering

Friday Jan 9

1.1, 1.2

N choose k and the Binomial Theorem, basics of a | b

Extra handout of the week: Using a pattern to design an inductive proof


Week 2

Day

Book Sections

Topics / Due Dates

Monday Jan 12

1.2

HW 1 due

 

GCD, Euclidean algorithm, more about linear combinations

Wednesday Jan 14

1.2

Relatively prime numbers, The key prime property about p | ab, Fundamental Theorem of Arithmetic

Friday Jan 16

1.2, A.3

Misc results about prime divisors, infinitely many primes, quick review of equivalence classes

Extra handout of the week: Factoring into powers of primes, and some useful calculations


Week 3

Day

Book Sections

Topics / Due Dates

Monday Jan 19

 

Martin Luther King Jr Day

No class

Wednesday Jan 21

1.3

HW 2 due

 

Definition of a == b (mod m), simplifying some calculations mod m, divisibility tests, squares mod m (part 1)

Friday Jan 23

1.3

Squares mod m (part 2), Fermat’s Little Theorem, Inverses mod m (part 1)

Extra handout of the week: Practice and results about inverses mod m


Week 4

Day

Book Sections

Topics / Due Dates

Monday Jan 26

1.3

Solving ax == b (mod m), Chinese Remainder Theorem

Wednesday Jan 28

1.4

HW 3 due

 

Z_m, well-defined functions, basic ring definitions

Friday Jan 30

1.4

Proofs of some ring laws (especially uniqueness results), units and zero divisors in Z_m

Extra handout of the week: Examples of non-commutative rings


Week 5

Day

Book Sections

Topics / Due Dates

Monday Feb 2

2.1

Ordered rings, proofs of some basic properties

Wednesday Feb 4

2.1

HW 4 due

 

Building Q out of Z, some remarks about Q[sqrt 2]

Friday Feb 6

 

Test 1 Review

Topic review for Test 1

Practice problems for Test 1: Partial solutions are now on eLC.

  • TEST 1 is Monday, February 9, in class. It’s a 55-minute test.
    • Expect tests to have about 5-7 questions.
      • One question asks you to repeat definitions.
      • There’s usually at least one question that makes you repeat a HW problem or class proof.
      • Tests are usually around half computation, half proof.
    • Just above the class schedule, there is now a link to a list of general study suggestions.

Week 6

Day

Book Sections

Topics / Due Dates

Monday Feb 9

 

TEST 1

Wednesday Feb 11

2.3

Complex numbers, conjugate, length |z|, polar form

Friday Feb 13

2.3

More polar form, DeMoivre’s Theorem, finding roots of complex numbers

Extra handout of the week: More practice with polar form and roots


Week 7

Day

Book Sections

Topics / Due Dates

Monday Feb 16

2.3, 2.4

E^(i*theta) notation, the Quadratic Formula for complex numbers

Wednesday Feb 18

2.4

HW 5 due

 

Cardano’s Cubic Formula(s)

Friday Feb 20

3.1

Polynomials R[x], basic degree results, long division

Extra handout of the week: When divisors go bad… counterexamples with polynomial division


Week 8

Day

Book Sections

Topics / Due Dates

Monday Feb 23

3.1

Root-Factor Theorem, GCDs of polynomials, associates in F[x]

Wednesday Feb 25

3.1

HW 6 due

 

Irreducible polynomials, partial fraction decomposition

Friday Feb 27

3.3

Rational Root Theorem, Gauss’s Lemma and using method of undetermined coefficients

Extra handout of the week: More details and practice with partial fractions

  • HW 7, due Wednesday March 4
  • TEST 2 is next Friday, March 6, right before Spring Break.
    • HW 7 solutions will have to be released a little sooner than usual.
    • Review materials should be uploaded by next Monday, March 2.

Week 9

Day

Book Sections

Topics / Due Dates

Monday Mar 2

3.3

Reducing polynomials to Z_p, Eisenstein’s Criterion

Wednesday Mar 4

 

HW 7 due

Solutions available soon on eLC… details coming

 

Test 2 Review

Friday Mar 6

 

TEST 2

Topic review for Test 2

Practice problems for Test 2: Solutions will be posted by Wednesday on eLC.

  • TEST 2 is Friday, March 6, right before Spring Break.
    • This test will be about the same length as Test 1.
    • There will be more emphasis on computation on this test, but you do have some proofs.

Week 10

Day

Book Sections

Topics / Due Dates

Monday Mar 16

5.1 (but also see first page of 3.2, maybe)

Motivation for F[alpha], vector space definition, subspaces and spans

Wednesday Mar 18

5.1

Linear independence, basis, dimension dim_F(V)

Friday Mar 20

5.1, 3.2

Formal definition of F[alpha], the “Degree Extension Formula”

Extra handout of the week: Linear maps of vector spaces

  • HW 8, due Monday March 23
  • This should be the only upcoming assignment due on Monday. Most of the questions should be possible with the material covered through Wednesday, but Friday should help give more examples and clarify the concepts. (It will have fewer tricky proofs.)

Week 11

Day

Book Sections

Topics / Due Dates

Monday Mar 23

3.2

HW 8 due

 

Definition of the minimal polynomial p_alpha(x), simplifying in F[alpha]

Wednesday Mar 25

3.2

Prove p_alpha(x) is irreducible, compute inverses in F[alpha], comparing extensions F[alpha] <= F[beta] (pt 1)

Friday Mar 27

3.2, 5.1

Iterated extensions F[alpha_1, …, alpha_n], splitting fields, comparing extensions (pt 2)

Extra handout of the week: Extra examples of iterated extensions

  • HW 9, due Wednesday April 1
    • The first seven problems will be covered this week. The last two problems will be covered in the first class next week.
    • The book has fairly limited coverage of iterated extensions F[alpha,beta]. This week’s handout should hopefully demonstrate the main ideas more clearly. If you want the handout released earlier than Friday, let me know.

Week 12

Day

Book Sections

Topics / Due Dates

Monday Mar 30

4.1

Ring homomorphisms, maps from F[alpha] to F[beta] (NOT IN BOOK)

Wednesday Apr 1

4.1

HW 9 due

 

Ideals, <a_1, …, a_n>

Friday Apr 3

4.1

Kernel of a morphism, principal ideal domains

Extra handout of the week: Comparing different ideals


Week 13

Day

Book Sections

Topics / Due Dates

Monday Apr 6

4.1

Congruence mod I, Quotients R/I

Wednesday Apr 8

4.2

HW 10 due

 

Isomorphisms, FHT*

Friday Apr 10

4.2

Practice with FHT and polynomials, Product rings R x S

*For more background / demonstration, see this handout.

Extra handout of the week: Morphisms to product rings


Week 14

Day

Book Sections

Topics / Due Dates

Monday Apr 13

4.2, 4.3

Harder morphisms with RxS, pictures of Z[i]*

Wednesday Apr 15

4.3

HW 11 due

 

Division and gcds in Z[i], some info about irreducible elements*

Friday Apr 17

 

Test 3 review

* The material from Monday and Wednesday will not be on Test 3, but parts of it may show up on the last HW or on the final exam.

Topic review for Test 3

Practice problems for Test 3: Solutions will be on eLC on Friday

  • Test 3 is next Monday, April 20.
    • It is the same length as the first two tests.
    • Weekend office hours: Sunday April 19, 2pm to 4pm, in connector between Boyd and Science Library

Weeks 15 and 16

Day

Book Sections

Topics / Due Dates

Monday Apr 20

 

TEST 3

Wednesday Apr 22

5.2

Basic constructions with compass and straightedge, constructible numbers

Friday Apr 24

5.2

Non-constructible numbers, constructible angles

Extra handout of the week: More background on constructible angles

  • HW 12, due Monday April 27 (last day of class)
    • Late HW cannot be accepted for this assignment. It should be completely graded, though, by early Tuesday April 28.
    • This HW’s first half deals with Z[i] from last week. If you have questions on it, try to get them addressed before the weekend.

 

  • The final exam takes place in our classroom, Boyd 303.
    • The 10:10am section has their test on Wednesday April 29 at 8am!
    • The 11:15am section has their test on Friday May 1 at 12pm.
    • I will be available mostly all day on Tuesday April 28 for last questions. (There probably won’t be any further practice problems.) Your HW should be graded and ready for pickup on that day.

Last updated: 4/24/15