Fall-2022. Math H113 (class # 31171). Honors Introduction to Abstract Algebra . TuTh 5:00-6:30, 70 Evans

Instructor: Alexander Givental

Office hours: Wed 2-4, in 701 Evans

Text: Lectures on Groups, Rings, and Fields which is my own E-book

Syllabus: the entire book.

Quizzes: weekly, in class (in the first 5 min), on Tue

Homework: weekly, due on Th, most likely via Gradescape

Grading policies:

Here is an ingenious (in my view) scheme of my own invention which I tried successfully in several courses and intend to use it this time. The starting point is: 30% weekly quizzes + 40% weekly hw + 30% final. However: each individual quiz or hw score which is below than or equal to (percentage-wise) your score on the final will be dropped - together with its weight! E.g.: if all your hw scores are below and all quizzes above your score on the final, then your total score is composed of 50% quizzes and 50% final. Thus, there are many reasons why you want to take quizzes and do hw (as well as many other exercises, not assigned as hw); yet a particular quiz/hw score can only improve your overall performance, but can never hurt your ultimate result compared to the final exam. In our time of many uncertainties this might be particularly useful: if for whatever reason you cannot come to a quiz or are not ready to submit a hw, you should not fret over this -- my policy does not penalize you for skipping it.

Besides, I don't have a preconceived distribution of As,Bs,..., and will be happy to give everyone an A should everyone learn the material well (for which I hope very much, especislly that the subject is interesting and simple). What exactly it means to learn well will become clear after some quizzes.

Thus, the general idea is that you don't compete with each other, but rather strive to learn the material the best you can. Respectively, collaboration and/or use of outside sources are not prohibited (tests excluded). Yet, each instance of this should be explicitly acknowledged in your homework. Failure to acknowledge one's use of somebody else's ideas is commonly known as academic plagiarism. So, let's practice the right ethics.


QZ1, on Tue, Aug. 30: on the material about functions from Lecture 1 of the textbook.
HW1, due Th, Sep. 1: Read Lectures 1 and 2. Solve Exercises 2,3,4,11.

QZ2, on Tue, Sep. 6: Read subsection D of Lecture 2 (about Euler's function) -- the quiz will be on this subject.
HW2, due Th, Sep. 8: Read Lectures 3 and 4. Solve exercises 14,15.

QZ3, on Tu, Sep. 13: on the material of lecture 5.
HW3, due Th, Sep. 15: Read Lectures 5 and 6. Solve exercises 19, 21, 40,47.

QZ4, on Tu, Sep. 20: on the cycle structure of permutations.
HW4, due Th, Sep. 22: Read Lectures 8 and 9. Solve exercises: 38, 50, 56, 57.

QZ5, on Tu, Sep. 27: on dihedral groups D_n (i.e. symmetry groups of regular n-gons)
HW5, due Th, Sep.29: Read lectures 10 and 11. Solve exercises 64, 66, as well as:
A: Prove that every finite group G is isomorphic to a subgroup of A_n for some n (depending on G).
B: Prove that all groups of order 45 are abelian.

QZ6, on Tu, Oct. 4: Direct products of groups.
HW6, due Th, Oct. 6: Read Lecture 12. Solve exercises 78, 86, 87. 91.

QZ7, on Tu, Oct. 11: Quaternions.
HW7, due Th, Oct. 13: Read Lectures 13, 14. Solve exercises 99, 119, 120 (take for granted the uniqueness of factorization in R[x,y] when R is a field), 121.

QZ8, on Tu, Oct. 18: Prime and maximal ideals.
HW8, due Th, Oct. 20: Read Lectures 14, 15. Solve exercises 126, 127, 128, 129.

QZ9, on Tu, Oct. 25: PIDs
HW9, due Th, Oct. 27: Read Lectures 16, 17. Solve exercises 130, 136, 139, 144.

QZ10, on Tu, Nov. 1: Waring's algorithm.
HW10, due Th, Nov. 3: Read Lectures 17, 18. Solve exercises 146, 149, 154, 160.

QZ11, on Tu, Nov. 8: Straightedge and compass constructions in elementary geometry (from Lecture 19)
HW11, due Th, Nov. 10: Read Lectures 18, 19. Solve excercises 162, 169, 170, 171.

QZ12, on Tu, Nov. 15: Multiple roots (read on your own Section A of Lecture 21).
HW12, due Th, Nov. 17: Read Lectures 20, 21. Solve exercises 172, 184, 188, 193.

No quiz on Tu, Nov. 22, nor HW due Th, Nov. 24

QZ13, on Tu, Nov. 29: Cyclotomic polynomials.
HW13, due TUESDAY, Nov. 29: Read Lectures 22, 23, 24. Solve exercises: 200, 201, 208, 214. (Don't neglect this hw, especially exercises 200, 201: they are your chance to understand Galois theory.)

Answers to HW

Sample final exam 1
Sample final exam 2