## 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.