A set of summary notes with definitions, theorems and exercises is available at the Google Classroom site, on the Classwork page, under the heading Materials.
There is no single required textbook for this course, but the following textbooks are recommended.
The first two of the following textbooks are the best, but only if they can be found for free or very inexpensively.
The third textbook is not quite as good as the first two, but is officially free.
The summary notes are keyed to the corresponding sections of these three texts.
Fraleigh, John, A First Course in Abstract Algebra, 7th ed., Addison- Wesley, Reading, MA, 2003;
Gallian, Joseph A., Contemporary Abstract Algebra, 7th ed., Brooks/Cole, Belmont, CA, 2010;
Urgent announcements may be sent out via campus email, so make sure you either check your Bard email regularly or have your Bard email forwarded to the email address of your choice.
Given the ever-changing situation with the Coronavirus, the following plans may be changed as circumstances require. The highest priority is to ensure the safety of everyone in the class.
Any changes to the way the class is conducted will be sent by email to the students in the class.
Class Meetings:
Class will meet in-person, and will also be available in video form for students who are not able to be in class.
All students are expected to complete the daily health screening using the Involvio, the College daily health screen app; a green pass on Involvio gives access to academic buildings.
Please do not attend class if you are sick, feel ill, know you may have been exposed to coronavirus, or have any of the symptoms listed on Involvio.
Class Meeting Expectations:
Masks must be worn throughout the class.
Food and drink will not be allowed.
Seating will be restricted to the fixed arrangement of chairs.
Attendance will be taken for the sake of contact tracing, but will not be a factor in grading.
To facilitate taking attendance, seating for the entire semester will be fixed according to where students sit on the first day of class.
Do not sit at the table in the center of the first row (it will be used for recording the class).
Remote Participation:
Every class will be recorded as a collection of videos, which will be posted at the class Google Classroom site.
Please watch the videos for each class before the next class meeting.
More Coronavirus Issues
Office Hours:
Office hours will be online, by appointment.
The method for signing up for appointments is found at the class Google Classroom site.
Homework:
Homework assignments will be posted at the class Google Classroom site.
Homework assignments must be submitted via the class Google Classroom site.
Each homework assignment must be uploaded as a single PDF file.
Exams:
The midterm exam will be in class, closed book.
The final exam will be take home, open book.
Work for the Course
Attendance:
It is expected that students attend all classes.
Homework:
Homework will be assigned every week, and will be posted at the class Google Classroom site. Submit the homework by the posted due date. Late assignments will not be accepted, except in genuine emergency situations.
All homework must be typed in LaTeX and must use the homework template of the Bard TeX Style file, which is available at the instructor's TeX website. If you need help with LaTeX, ask the instructor.
You are encouraged to work with other students in solving the homework problems. However, for the sake of better learning, as well as honesty, please adhere to the following guidelines:
Write up your solutions yourself.
Acknowledge in writing anyone with whom you work and any assistance you receive.
Acknowledge in writing any revisions of your work based upon solutions given in class.
Failure to indicate collaboration, assistance or sources will be construed as plagiarism.
The use of homework solutions found on the web or elsewhere will be treated similarly to plagiarism on exams.
Your solutions should be written clearly and carefully, as described below.
Exams:
Each exam will be as follows
Midterm Exam (In-Class): Thur., Oct. 22
Final Exam (Take-Home): Tue., Dec. 15
What is Math 332
This course is an introduction to modern abstract algebraic systems, specifically groups, rings and fields. The focus of the course is a rigorous treatment of the basic theory of groups (subgroups, quotient groups, homomorphisms, isomorphisms, group actions), and an introduction to rings and fields (ideals, polynomials, factorization).
This course provides an opportunity for students to develop their skills at formulating and writing rigorous mathematical proofs, and it makes use of the methods and concepts of Proofs & Fundamentals (Math 261).
The prerequisite for this course is Proofs & Fundamentals (Math 261) or permission of the instructor.
Office Hours
If you have any problems with the course, or any questions about the material, the assignments, the exams or anything else, please see the instructor about it as soon as possible.
Due to Coronavirus, all office hours this semester will be online, by appointment. Standard appointment slots are available on Mondays and Fridays. The simplest way to make an appointment for office hours is to use Google Calendar Appointment Slots, where available 15 minute time slots are listed. To schedule office hours at other times, email the instructor,
To begin with, please sign up for a single 15 minute time slot at a time, to make sure that there are enough time slots for everyone, but if 15 minutes are not enough time, then sign up for additional time.
Try to sign up ahead of time (preferably a day before the scheduled meeting).
Make sure to write your name when you sign up for a time slot; if a group of students wants to have office hours together, make sure to write all your names.
You will receive an email invitation to to a Zoom meeting at the time of your appointment.
Calculators, Computers and Electronic Devices
Use of a computer will be needed for typing the homework in LaTeX, which will be required for all homework assignments, as discussed in class.
Calculators are not needed for this class.
Electronic devices, including cell phones, tablets and laptop computers, may be used during class only for reasons related to the class, for example as calculators, to take notes or to read the text.
Texting, messaging and using social media is not allowed during class.
Mathematics Study Room
The Mathematics Study Room is staffed by undergraduate mathematics majors, some of whom will be able to assist you with the material for this course; you can go to the study room to work on your homework and ask for help with the material
Hours: TBA Location: TBA
Tutor
There is no dedicated tutor for this course. If you need additional help beyond office hours, some of the tutors in the Mathematics Study Room can assist you.
Grading
Grades will be determined roughly 50% by the homework assignments and 50% by the exams. Class participation will be taken into account positively, especially in cases of borderline grades.
Grades will be determined by work completed during the semester, except in cases of medical or personal emergency. There will be no opportunity to do extra credit work after the semester ends.
This course is graded using letter grades. During fall 2020, if you want to take the course Pass/Fail, you must submit a request to do so to the Registrar's Office at any time prior to the start of the spring 2021 semester.
Accommodations
Students with documented learning and/or other disabilities are entitled to receive reasonable classroom and testing accommodations. If you need accommodations, please do the following.
Contact the instructor at least one week prior to each exam, quiz or other instance of accommodation, to arrange appropriate scheduling.
If you feel comfortable doing so, discuss your accommodations with the instructor at the beginning of the semester.
If you need to miss a class for any reason (sports team, religious holiday, etc.), it is your responsibility to contact the instructor and find out about the material and assignments you missed.
Travel plans that do not take into account the dates of quizzes and exams are not a valid reason to miss an exam; there is no guarantee that quizzes and exams will be available early to accommodate travel plans.
Everyone makes honest mathematical mistakes, but there is no reason to get in your own way by writing your proofs with incomplete sentences and other grammatical mistakes, by using undefined symbols for "variables" or by engaging in other forms of sloppy writing. The goal of writing mathematics is two-fold: making sure that a proof is correct in all details, and communicating the proof so that others can understand it. To help achieve those aims, mathematics must be written carefully, and with proper grammar, no differently from any other writing. Properly written proofs entail the following basic points.
Write your homework assignments neatly and clearly.
Use correct grammar, including full sentences and proper punctuation.
Justify each step in a proof, citing the appropriate results from the class notes as needed.
Use definitions precisely as stated.
Be very careful with quantifiers.
Strategize the outline of a proof before working out the details; the outline of a proof is always determined by what is being proved, not by what is known.
Distinguish between scratch work and the actual proof; scratch work can be in any order, but the actual proof always starts with what is known and deduces the desired result.
Proofs should stand on their own; check your proofs by reading them as if they were written by someone else.
Please see the instructor if you have questions about writing – or doing – the homework assignments.
Learning to Do Rigorous Mathematics
Proofs-based mathematics courses are very different from computation-based mathematics courses such as Calculus. The ways you studied, did homework and took exams in computation-based courses was appropriate for those courses, but not for proofs-based courses. Approach proofs-based courses with the idea that you will be doing things differently from what you did in computation-based mathematics courses.
The material in this course is much more abstract, and requires much more precision in both studying and problem solving, than the material you saw in courses such as Calculus. For some students, a proofs-based course such as this one is the first time that they found a mathematics course really challenging, which can be intimidating at first, but is in fact completely normal. Everyone, including the very best mathematicians, reaches a level of mathematics that he or she finds difficult; what varies from person to person is only what that level is. If you made it this far in mathematics and you only now first encounter substantial difficulty in learning the material, you are doing fine.
In general, the more advanced you get in mathematics (or any subject), the larger the percentage of learning that takes place outside of class, including from the textbook, from other sources, from office hours, from tutors and from your fellow students (not necessarily in that order).
In Calculus courses, where the material can mostly be learned in class, reading the textbook is not necessarily very important. By contrast, in proofs-based courses reading the textbook carefully, and seeking help with those parts of the textbooks that you find difficult, is crucial.
In proofs-based courses, reading the textbook is very different from reading fiction, in two ways. First, reading proofs-based mathematics, which cannot be done without pencil and paper in hand, requires active engagement by regularly stopping to work out the details of what is written. Make sure you know why each step in a proof is true before moving on, and if you are unable to figure out one or more steps of a proof, seek help. Second, mathematics is not read in order from beginning to end, but "from the outside in." When you read a proof, start by looking for the overall idea of the proof, and then figure out the strategy that is being used, and only then go through the details one step at a time.
In Calculus courses, solutions to homework exercises are usually written as a collections of equations, with little or no words explaining the solution. By contrast, rigorous proofs are, fundamentally, convincing arguments, and to make a good argument, words are needed to direct the logical flow of the ideas; to explain what is assumed and what is to be proved; and to state what previous results are used. In particular, rigorous proofs are written using full sentences, and with correct grammar and punctuation, because doing so helps make the arguments more clear and precise.
In Calculus courses, solutions to homework problems are usually written directly, with little revision. By contrast, rigorous proofs should be written the same way a paper in a humanities course is written, by first making an outline (often called "strategizing a proof"); then sketching out a rough draft; then revising the draft repeatedly until the proof works; and, lastly, writing the final draft carefully and typing it in LaTeX.
Revising a draft of a rigorous proof should be done exactly as revising a draft of a paper in a humanities course, which is to read it as if you are not the author, but rather as if you are someone else in the class, and making sure that each sentence makes sense as written, without recourse to unwritten explanations.
Learning to write rigorous proofs takes time, and you should not expect to master it instantly.
A very good way to improve your skill at writing proofs, and to do as well as possible on the homework in this course, is to bring a draft of every homework assignment to office hours before you write up and submit the final draft.
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