Math 113: Introduction to Abstract Algebra

Instructor: David Corwin (dcorwin at berkeley dot edu). In all e-mail correspondence, please include "[Math113]" in the subject line.

GSI: Julian Chaidez (jchaidez at berkeley dot edu)

Lecture: MW 5pm-6:30pm in Hearst Mining 310

GSI Office hours: GSI Office hours are tentatively in Evans 961. They are 3:30-6 MTuTh, 3-4 W and 3:30-5 F.

My Office hours: Location: Evans 749, Monday 2-3pm, Tuesday 3:45-5:45pm. Subject to change if I am away: No office hours April 30. Extra office hours 1-2pm on April 29 and 3:45-4:45pm on May 2.

Final office hours or review: Wednesday, May 8.

Final exam: Friday, May 17, 2018 3-6pm. Location HM 310 (same as class).

Prerequisites:

Text: The primary text for this course is Notes on Abstract Algebra by Alexander Paulin [P]. Students should feel free to consult other books for additional exercises and/or alternative presentations of the material. Wikipedia also has lots of great articles on the topics at hand (e.g., see articles on Abstract Algebra, Groups, Rings, Fields, and many of the other concepts we will discuss in class).Students are expected to read the relevant sections of the textbook, as the lectures are meant to complement the textbook, not replace it, and we have a lot of material to cover.

Grading: 30% homework, 70% exams. The lowest two homework scores will be dropped. No makeups for the midterm will be given except in cases requiring special accommodation. The midterm will be on March 20, in class. Your exam grade, which counts for 70%, will be computed based on the maximum of the following two schemes: 0.25*midterm+0.75*final and 0.35*midterm+0.65*final

Website: For now, the only website is this page, http://math.berkeley.edu/~dcorwin/math113.html. I will use bcourses for solutions and other non-public information, such as my phone number.

Course policies:

• Homework will be assigned regularly (see the syllabus) and due at the beginning of class (by hand or e-mail). I grant extensions in reasonable circumstances, but you must talk to me as early as possible. The longer you wait, the less flexible I will be.
• You may work together to figure out homework problems, but you must write up your solutions in your own words in order to receive credit. In particular, please do not copy answers from the internet or solution manuals. Since a major purpose of the homework is to prepare you for the exams, I encourage you to give each problem an honest shot by yourself (say, at least thirty minutes) before discussing it with others. Another useful practice is if you're stuck on a problem, come to office hours and ask for a hint. The more you figure out on your own, the better your understanding of the material, and the better you'll do both on the exams and in your future endeavours that might require abstract algebra.
• The usual expectations and procedures for academic integrity at UC Berkeley apply. Cheating on an exam will result in a failing grade and will be reported to the University Office of Student Conduction. Please don't put me through this.
• Please let me know sooner rather than later if you need any accommodations related to the Disabled Student Program (DSP). I am more than happy to make arrangements, but it really helps if you tell me earlier rather than later.
• Per university guidelines, it is your responsibility to notify the instructor in writing by the end of the second week of classes (February 1) of any scheduling conflicts due to religious observance or extracurricular activities, and to propose a resolution for those conflicts.

Additional resources:

• Abstract Algebra: Theory and Applications by Thomas Judson
• Abstract Algebra by John Fraleigh

Course Overview: Outlined below is the rough course schedule. Depending on how the class progresses it may be subject to minor changes over the course of the semester.

Course calendar; "[P] refers to Paulin's notes, [J] refers to Judson's book, and [W] to Wikipedia.

Date	Meeting	Topics	References	Comments	Alternative References
1/23	1	Logistics/Syllabus, Overview of Course	Section 1.1 of [P]
1/28	2	Sets and Functions	Section 1.2-1.3 of [P], plus Notes on mathematical proofs - PLEASE READ		Chapter 1 of [J],
1/30	3	Integers and Modular Arithmetic	Section 2 of [P], see these problems for optional practice.		2.2 and 3.1 of [J], GCD, Euclidean Division and Modular Arithmetic on [W]
2/4	4	Groups and Homomorphisms	Section 3.1 of [P], see these problems for optional practice.		3.2 and 9.1 of [J], Groups, Group Homomorphism on [W]
2/6	5	Subgroups, Cosets, and Lagrange's Theorem	Section 3.2 of [P], see these problems for optional practice.	HW 1 Due	3.3, 6.1, 6.2 of [J], Subgroup, Coset, Lagrange's Theorem on [W]
2/11	6	Finitely Generated Groups	Section 3.3 of [P], see these problems for optional practice.		4.1 of [J], Order, Generating Set, Cyclic Groups on [W]
2/13	7	Permutation Groups and Group Actions	Section 3.4 of [P], see these problems for optional practice.	HW 2 Due	5.1, 14.1 of [J] Permutation Group, Group Action on [W] (If you look at [W], note that all our group actions are left group actions.)
2/18		HOLIDAY
2/20	8	The Orbit-Stabiliser Theorem and Sylow's Theorem	Section 3.5 of [P], see these problems for optional practice.	HW 3 Due	14.1, 14.2, 15.1 of [J]
2/25	9	Quaternion Group, Begin Symmetric Groups	See the Mathworld article on the quaternion group or the group wiki article. Note that "conjugacy class" is the same as "orbit under conjugation."		See Example 1.12 of these notes.
2/27	10	Finite Symmetric Groups	Section 3.6 of [P], see these problems for optional practice.	HW 4 Due
3/4	11	Symmetry of Sets with Extra Structure	Section 3.7 of [P], see these problems for optional practice.
3/6	12	Normal Subgroups and the Isomorphism Theorems	Section 3.8 of [P], see these problems for optional practice.	HW 5 Due	Chapter 10 of [J]
3/11	13	Direct Products and Direct Sums	Section 3.9 of [P], Notes on centralizers		9.2 of [J]
3/13	14	Finitely Generated Abelian Groups	Section 3.10 of [P]
3/18	15	Finite Abelian Groups	Section 3.11 of [P]	HW 6 Due	13.1 of [J]
3/20	16	Midterm Exam
3/25-3/29		Spring Break
4/1	16	Rings and Field: Basic Definitions	Section 4.1 of [P]		16.1, 16.2 of [J]
4/3	17	Subrings, Ideals, and Homomorphisms	Section 4.2, 4.3 of [P]	HW 7 Due	16.3 of [J]
4/8	18	Polynomial Rings	4.4 of [P]		17.1 of [J]
4/10	19	Field of Fractions	4.5 of [P]	HW 8 Due	18.1 of [J]
4/15	20	Characteristic	4.6 of [P]		16.2 of [J]
4/17	21	Ring Extensions, begin Principal Ideals	4.7 of [P] and handwritten notes on Bcourses (4.7 is sparse), begin 4.8 of [P]	HW 9 Due
4/22	22	Principal, Prime, and Maximal Ideals	4.8 of [P]		16.4 of [J]
4/24	23	Factorization in Integral Domains and UFD's	up to p.64, also my notes on factorization in rings	HW 10 Due
4/29	24	A little bit about PID's, then factorization of polynomials	my notes (see link under 4/24)
5/1	25	More on polynomials, overview of Galois Theory and Algebraic Geometry	my notes (see link under 4/24)
5/17	1	Final Exam

Homework and Exams:

• Homework 1, due Wednesday, February 6 (on Gradescope).
• Homework 2, due Wednesday, February 13 (on Gradescope).
• Homework 3, due Wednesday, February 20 (on Gradescope).
• Homework 4, due Wednesday, February 27 (on Gradescope).
• Homework 5, due Wednesday, March 6 (on Gradescope) (will give extension for those who couldn't access homework).
• Homework 6, due Monday, March 18 (on Gradescope).
• Practice Midterm 1 (ignore question 7), Solutions.
• Practice Midterm 2 (ignore question 7), Solutions.
• Practice Midterm 3 (ignore question 7), Solutions.
• Practice Midterm 4 (ignore question 7) with solutions.
• Homework 7, due Wednesday, April 3 (on Gradescope).
• Homework 8, due Wednesday, April 10 (on Gradescope).
• Homework 9, due Wednesday, April 17 (on Gradescope).
• Homework 10, due Wednesday, April 24 (on Gradescope).
• Homework 11, due Saturday, May 4 (on Gradescope).
• Practice Final 1 (ignore question 7b,c), Solutions.
• Practice Final 2 (ignore question 5,7, for 6 assume no rational roots), Solutions.
• Practice Final 3 (ignore question 6), Solutions.