Math 100B: Abstract Algebra II (Winter 2020)

The course meets MWF 11-11:50am in Peterson 102. For the course syllabus and policies, see the bottom of this page; however, note that this material is subject to revision until the first lecture.

Announcements (most recent first)

• Review session: Friday 4-6pm, HSS 1128A (Alex).
• Special office hours during Week 10 (these supersede the usual schedule):
  • Tuesday 3-5pm (Alex)
  • Wednesday 3:15 - 5:15pm (Alina)
  • Thursday 3:30-5:30pm (Alex)
  • Sunday 3-6pm (Alina)
• Practice problems for the final exam.
• The Budapest Semesters in Mathematics are a wonderful opportunity to explore more advanced mathematics. The semester program applications are due April 1 and they require students to have taken a course in abstract algebra or real analysis.
• Wednesday Feb 26 my office hours will be 9:00-10:45am (instead of the usual 3:15-5:15pm).
• Alex will hold extra office hours Friday Feb 21 2-4pm.
• This week my office hours will by Tuesday (Feb 11) 6:30-8pm (instead of Wednesday).
• I have posted practice problems for the midterm.
• This week my office hours will by Tuesday (Jan 28) 4-5:30pm (instead of Wednesday).
• I have posted practice problems for the quiz and the proof of the first isomorphism theorem.
• Since Wednesday Jan 29 is the quiz, HW3 will be due on Wednesday Feb 5. I suggest you go through the problems in sections 3.3 and 3.4 from HW3 as you prepare for the quiz. The quiz itself will cover the material from weeks 1-3; namely, sections 3.1-3.4, as well as the material on polynomial rings which corresponds to a subset of section 4.1. CRT will not be part of the quiz.
• The textbook for MATH 100C will be Algebra by Michael Artin.
• My office hours on Wednesday 1/15 will take place virtually. We will use zoom. Please use this link to connect: https://ucsd.zoom.us/j/517486554. You might need to install zoom to be able to connect.
• There will be no lectures and no discussion sections on the following university holidays: Monday January 20 (Martin Luther King Day); Monday February 17 (Presidents Day).

Homework assignments

All numbering refers to Nicholson.


• HW1 (due W 1/15)
  • 3.1: 6, 16, 17, 28
  • 3.2: 1, 4, 10
  • 4.1: 1, 3
  In problem 16(a), section 3.1, Z(R) is supposed to be the center of the ring. The book has a typo where the Z looks like the notation for the ring of integers, but they mean the center of the ring as defined in Example 12.


• HW2 (due W 1/22): solutions to selected problems
  • 3.2: 2, 19, 20, 21, 25, 30
  • 3.3: 4, 5, 9


• HW3 (due W 2/5) solutions to selected problems
  • 3.3: 12, 20, 23, 32
  • 3.4: 1, 16, 32, 33, 44
  • 4.1: 13, 23, 24, 26, 30


• HW4 (due F 2/14 in lecture) solutions to selected problems
  • 4.2: 3, 4, 10, 22, 28, 43
  • 4.3: 1, 10, 11, 29


• HW5 (due W 2/26) solutions to selected problems
  • 4.1: 25
  • 5.1: 3, 4, 7, 10, 12
  • 5.2: 1, 8, 11, 13, 14, 16


• HW6 (due W 3/4) solutions to selected problems
  • 6.1: 1, 2, 9, 26, 31
  • 6.2: 1, 4, 7, 13


• HW7 (due F 3/13 in lecture) solutions to selected problems
  • 6.3: 1, 6, 11, 13
  • 6.4: 2, 3, 5, 8, 21, 22

Topics by date

All numbering refers to Nicholson. Listings for future dates are subject to adjustment based on how far we get in class.

• M 1/6: rings, fields, polynomials, homomorphisms of rings, isomorphisms of rings (3.1, 3.4, 4.1)
• W 1/8: subrings, endomorphisms, automorphisms, characteristic of a ring (3.1, 3.2)
• F 1/10: subfields, product of rings, integral domains, center of a ring (3.2, 3.3)
• M 1/13: field of fractions of an integral domain, Gaussian integers, quadratic fields (3.2)
• W 1/15: ideals and factor rings (3.3)
• F 1/17: maximal ideals, prime ideals, ideals in the factor ring, simple rings (3.3)
• W 1/22: Ring homomorphisms, kernel, image, isomorphism theorems (3.4)
• F 1/24: Isomorphism theorems, the Chinese Remainder Theorem (3.4)
• M 1/27: The Chinese Remainder Theorem, the first isomorphism theorem (3.4), quiz review
• W 1/29: Quiz
• F 1/31: Polynomials, long division, divisibility (4.1, 4.2)
• M 2/3: Long division over a field, repeated roots, divisibility, irreducibles (4.2)
• W 2/5: The greatest common divisor for polynomials (4.2)
• F 2/7: Quotients of polynomial rings (4.3)
• M 2/10: Polynomials with integer coefficients; the ring of polynomials over a ring (4.1, 4.2)
• W 2/12: Midterm
• F 2/14: Prime and irreducible elements, euclidean domains (5.1)
• W 2/19: Prime and irreducible elements, principal ideal domains, unique factorization domains (5.1, 5.2)
• F 2/21: PIDs are UFDs (5.1)
• M 2/24: Factorization in Z[X]; rings of polynomials over a UFD is a UFD (5.1, 5.2)
• W 2/26: Factorization in Z[X]. Irreducibility criteria (e.g. Eisenstein)
• F 2/28: Vector spaces over a field (6.1)
• M 3/2: Bases, dimension, span, linear independence in vector spaces. Field extension, degree of a field extention, algebraic and transcendental elements(6.1, 6.2)
• W 3/4: Minimal polynomial of an algebraic element, degree of an extension (6.2)
• F 3/6: Splitting fields (6.3)
• M 3/9: Algebraic closure, factoring of polymomials in algebraic extensions (6.3, 6.4)
• W 3/11: Finite fields, automorphisms and field extensions (6.4)
• F 3/13: Primitive elements, the multipicative group of a finite field is cyclic (6.4)

Course syllabus

Math 100A/B/C is a rigorous three-quarter introduction to the methods and basic structures of higher algebra. 100B will focus on rings and fields. Topics include: linear algebra over fields; rings; polynomial rings; ideals and quotients; unique factorization; quadratic number fields; linear algebra over rings.

UCSD also offers a two-quarter algebra sequence, Math 103A/B (offered both fall/winter and winter/spring). Between the two, Math 100 offers a greater emphasis on concepts and mathematical rigor, as well as some advanced topics not covered in Math 103 (e.g., Galois theory). Math 100 is recommended for students planning further study in pure mathematics, while Math 103 is recommended for most other students. Students may not receive credit for both Math 100B and Math 103B. (Note that 100A is a valid prerequisite for 103B, but 103A is not a valid prerequisite for 100B.)

Instructor: Alina Bucur
Office: AP&M 7151
Email: alina@math.ucsd.edu
Office hours: see above or by appointment.
I reserve the right to leave after 4:30pm if no one is present or has warned me in advance that they plan to come late. Exceptions (including adjustments for exams) will be noted in the course announcements.

TA: Alex Mathers
Office: AP&M 5412
Email: amathers@ucsd.edu
Office hours: see above

Lectures: MWF 11:00-11:50am in Peterson 102. No lectures on the following university holidays: Monday, January 20 (Martin Luther King Day); Monday, February 17 (Presidents Day).

Sections: M 6:00-6:50pm, 7-7:50pm in APM 5402. No sections on the following university holidays: Monday, January 20 (Martin Luther King Day); Monday, February 17 (Presidents Day).

Text: Abstract Algebra by W. Keith Nicholson, fourth edition (required). This is the same text that was used for the two 100A lectures in the fall (which might make it easier to find a used copy). The material for 100B will be drawn primarily from chapters 3-8.

Prerequisites: Math 100A or consent of instructor. Any flavor of Math 100A is accepted, including the two fall 2019 lectures taught by Dragos Oprea and Alireza Salehi Golsefidy, or any previous year's edition. However, Math 103A is not accepted as a prerequisite.

Homework: Weekly assignments. In general, problem sets will be posted after Monday's lecture, to be due on Wednesday. To receive credit, homework must be submitted in the dropbox in the basement of APM no later than 9pm on the due date. (Fair warning: the building automatically locks at 9:20pm.) No extensions will be granted; see below for grading policies.

Quiz: in class on Wednesday, January 29.
Midterm: in class on Wednesday, February 12. (It will include all material since the start of the term.)
No makeup exam will be given; see below for exam policies.

Final exam: Monday, March 16, 11:30am-2:30pm, location TBA. Please note that by signing up for this course, you are agreeing to sit for the final examination at this date and time. See UCSD exam policies as well as course-specific policies below.

Grading: 25% homework, 10% quiz, 25% midterm, 40% final exam; or 30% homework, 30% midterm, 40% final exam; or 40% homework, 10% quiz, 50% final exam (whichever is higher). In all formulas the lowest homework assignment will be dropped.

The conversion of raw percentages into letter grades will be made in order to maintain a grade distribution comparable with historical averages for this course. However, the following minima are guaranteed:

Percentage	97	93	90	87	83	80	77	73	70
Minimum grade	A+	A	A-	B+	B	B-	C+	C	C-

Additionally, any score in at least the 85th percentile or higher is guaranteed at least an A-, while any score in the 70th percentile or higher is guaranteed at least a B-.

Notwithstanding the above, to receive a passing grade, you must fulfill the following conditions.

• You must take the final exam, at the scheduled time and place (unless granted a mandated accomodation; see policies), and receive a passing grade.
• You must not be found in violation of UCSD's academic integrity or harassment policies.

Please access TritonEd for homework and exam scores. No other material will be posted there.
Keep all of your returned exams and homeworks. If there is any mistake in the recording on TritonED of your scores, you will need the original assignment/quiz in order for us to make a change. The error has to be reported within 1 week since it occurred. No error reports will be accepted after week #9 of the term.

Electronic devices: Please do not use devices (such as cell phones, laptops, tablets, iPods) for non-class-related matters while in class/section. No visual or audio recording is allowed in class/section without prior permission of the instructor (whether by camera, cell phone, or other means).

Policies

No extensions will be given for homework assignments, but the lowest homework assignment will be dropped. This applies even if you add the course late (as of the first day of classes, there was no waitlist).

At the top of each homework assignment, you must specify all outside resources that you consulted, or write "None" if none were used. You do not need to report use of the main textbook, any additional notes distributed via this web site, your own notes from lecture, or consultations with the professor or TA (including discussions during sections or office hours). You do need to report use of any other textbooks, any materials found online (in a precise fashion; e.g., for Wikipedia you must specify particular articles), and any consultation with anyone other than the professor or TA (including study group partners). If you collaborate with other students in the class, you must write up your solutions in your own words; copying solutions verbatim from another student, or quoting another source without attribution, will be treated as a violation of academic integrity (see below).

All exams will be closed-book: no outside materials may be consulted. This includes the textbook, lecture notes, the Internet, and anyone other than the exam proctor. We reserve the right to:

• require students to produce their UCSD student ID cards for admission to the exam room and/or submission of completed exams;
• assign seating before or during the exam;
• make video recordings for the purposes of monitoring academic integrity.

Exam accommodations will be made only in the following cases mandated by university policies. (Other circumstances, such as a family/medical emergency during finals week, may be covered by the incomplete policy; see below.)

• For disability accommodations, please follow Department of Mathematics procedures. Having documentation on file with OSD is not sufficient.
• For athletic accommodations, please have a cognizant representative of the Department of Athletics contact the instructor.
• For religious accommodations consistent with UCSD policy, please contact the instructor.

All accommodation requests must be made with sufficient advance notice, preferably by the end of week 1. No accommodations are available for homework assignments.

No makeup exams will be given. A missed exam will be scored 0 and handled in accordance with the course grading scheme (see above).

A request for an Incomplete grade will only be granted in accordance with UCSD policies. In particular, you must be on track to receive a passing grade based on your submitted homework, quiz and midterm results (using the 25/10/25/40 option). To convert an incomplete into a final grade, you must provide to the instructor proper documentation of the circumstances leading to the Incomplete, and arrange with the instructor to complete all outstanding course requirements no later than the end of the subsequent quarter.

Violations of UCSD's academic integrity policies (cheating, plagiarism, etc.) will be handled by the instructor using UCSD administrative measures.
Any violation of UCSD's academic integrity or harassment policies and will result in failing the class. If you suspect a violation, please bring it to the attention of the instructor and/or TA immediately.