## Spring-2022. Math H113. Honors Introduction to Abstract Algebra . TuTh 12:30-2:00, 2 Evans

** Instructor:** Alexander Givental

** Office hours: ** Th 3-4 in 701 Evans, and Mon 11:30-12:30 on Zoom

** Text: ** The whole book
* Lectures on Groups,
Rings, and Fields* by Alexander Givental

Also, * Topics in Algebra * by I. N. Herstein
2nd edition can be recommended (but not required, and will not be really
followed).

** Grading:** Final 30%, quizzes 30%, hw 40% (these details are
still to be discussed and might be subject to change because of COVID constraints)

More precisely: Each individual hw or quiz scores below the score on the final will be dropped (together with their share in the total percentage), and the relative weight of hw / quizzes / final recalculated accordingly.

** Homework: ** To be posted to this website weekly, due on Tue in class
or by email before the start of the lecture

** Quizzes:** Typically 1 question for 5 minutes at the start of each Thursday
lecture (but not during the first two weeks, when we are on zoom)

** HW1, due Tue, Jan. 25:** Read Lectures 1 and 2. Solve Exercises 2,4,5,9. Answers to HW1

** HW2, due Tue, Feb. 1:** Read Lectures 3 and 4. Solve exercises 12,13.
Answers to HW2

** QZ1, on Th, Feb. 3:** At the start of the lecture we will have a
5-min/ 1-question quiz * on the material of Lecture 4. *

** HW3, due Tue, Feb. 8:** Read Lectures 5 and 6. Solve exercises 17, 28, 30,33. Answers to HW3

** QZ2, on Th, Feb. 10:** on the material of lecture 5.

** HW4, due Tue, Feb. 15:** Read Lectures 7 and 8. Solve exercises 42, 45, 47, 51. Answers to HW4

** QZ3, on Th, Feb. 17:** Cycle structure of permutations.

** HW5, due Tue, Feb. 22:** Read Lectures 9 and 10. Solve exercises 58, 60, 62, 65. Answers to HW5

** QZ4, on Th, Feb. 24:** Direct products of groups.

** HW6, due Tue, Mar. 1:** Read Lectures 11 and 12. Solve exercises 72, 80, 81, 82. Answers to HW6

** QZ5, on Th, Mar. 3:** Quaternions.

** HW7, due Tue, Mar. 8:** Read Lectures 13 and 14. Solve exercises 95, 110,
111 (take for granted that in polynomial rings, factorization into irredcibles is unique), 112. Answers to HW7

** QZ6, on Th, Mar. 10:** Euclidean rings

** HW8, due Tue, Mar. 15:** Read Lectures 15, 16. Solve exercises 118, 121, 127, 130. Answers to HW8

** QZ7, on Th, Mar. 17:** Gaussian integers

** HW9, due Tue, Mar. 29:** Read Lecture 17. Solve exercises 135, 138, 143, 145. Answers to HW9

**QZ8, on Th, Mar. 31:** Waring's algorithm.

**HW10, due Tue, Apr. 5:** Read Lectures 18, 19. Solve exercises 149, 151, 158, 159. Answers to HW10

**QZ9, on Th, Apr. 7:** Straightedge and compass constructions.

**HW11, due Tue, Apr. 12:** Read lectures 20, 21. Solve exercises 160, 161, 173, 177. Answers to HW11

**QZ10, on Th, Apr. 14:** Multiple roots.

**HW12, due Tue, Apr. 19:** Read lectures 22, 23. Solve exercises 180, 182,
189, 190. (Beware: this hw is volumous but important; don't neglect it ---
this is your chance to understand Galois theory.) Answers to HW12

**QZ11, on Th, Apr. 21:** Cyclotomic polynomials. CANCELLED

**HW13, due Tue, Apr. 26:** Read lectures 24, 25, and Appendix II.
Solve exercises: 197, 198, 202, 203. Answers to HW13

**QZ12, on Th, Apr. 28:** Solvable groups. CANCELLED

**HW14, due Tue, May 3:** Read lectures 26, 27. Solve exercises: 204, 205,
211, 213. Answers to HW14

On the history of Cardano's formula (discussed in lecture 27) I highly
recommend the following (truly spectacular) presentation by
* Veritasium *

Sample final exam 1

Sample final exam 2

Those who want to refresh their knowledge of complex numbers and/or determinants
are welcome to read (among many other available sources)
Section 2 of Chapter 1 and Section 3 of Chapter 2 respectively of my own book
* Linear Algebra * (which, were the author not seeking eternal fame,
could have been titled "Linear Algebra Done Wrong") currently accessible online
at my
Math H110
course website.