MATH GU4041 Introduction to
Modern Algebra I
Fall 2025
Time and place: MW 1:10--2:25, location 312 Mathematics.
Instructor: Robert Friedman (x4-4355). Office: 605 Mathematics.
Office hours: Until further notice, my office hours are Tuesdays 1--2 PM and Thursdays 2--3 PM in 605 Mathematics, but feel free to email me if you need to set up another time, either in person or on Zoom.
Email: rf@math.columbia.edu
Teaching Assistants: TBA1tba@columbia.edu, TBA2 tba1@math.columbia.edu Office hours TBA.
Goal of the course: This is the first semester of a two-semester sequence on Abstract Algebra. This semester will concentrate on group theory. Beyond covering the essentials of group theory, the course aims to develop your ability to work with abstract mathematical concepts.
Prerequisites: Math UN1202 (Multivariable Calculus) and Math UN2010 (Linear Algebra), or equivalent courses, are prerequisites for this course. You should also be familiar with complex numbers, mathematical induction and other methods of proof, and in general have a certain confidence in your abilities to handle abstract mathematical reasoning. A prior course which involves writing proofs such as Honors Math A/B or Introduction to Higher Mathematics is strongly recommended. If this is your first proof based course, you should consider attending the Introduction to Proofs Workshop Math UN2005.
Text: There is no required text. Course notes will be posted.
Recommended texts: There are very many books covering the basics of Abstract Algebra; browsing the library or the internet is recommended for further examples, history, or different approaches to the material. Here is a selection of some recommended ones.
Michael Artin, Algebra (Second Edition), Prentice-Hall 2011. ISBN-13: 978-01324137-0. Most of what we will cover this semester can be found in Chapters 2, 6, and 7.
D. Dummit and R. Foote, Abstract Algebra, (Third edition), John Wiley and Sons, 2004. ISBN-13: 978-0471433347. Most of what we will cover this semester can be found in Chapters 1 through 5.
John Fraleigh, A First Course in Abstract Algebra (Seventh Edition), Addison Wesley 2002. ISBN-13: 978-0201763904. Most of what we will cover this semester can be found in Sections 1 through 17 and 34 through 37.
Joseph Gallian, Contemporary Abstract Algebra (Ninth Edition), Cengage Learning 2016. ISBN-13: 978-1305657960
Thomas Hungerford, Abstract Algebra: An Introduction (Second Edition), Brooks Cole 1996. ISBN-13: 978-0030105593
I. Herstein, Abstract Algebra, John Wiley 1996. ISBN-13: 978-0471368793
T. Judson, Abstract Algebra: Theory and Applications. There is a free online edition available here, with instructions on how to purchase a hard copy.
S. Lang, Undergraduate Algebra (Third Edition), Springer 2005. ISBN: 0-387220259
Of these, the books by Fraleigh, Gallian, and Judson are the most elementary, the book by Artin is at an intermediate level, and the book by Dummit and Foote is the most advanced.
Homework: There will be weekly problem sets, due on Mondays, and typically posted after class on the previous Wednesday. Problems will be assigned from the posted notes. The first problem set will be due on Monday, September 15. You should attempt every homework problem and eventually understand how to do every problem correctly. Collaboration and discussion with your classmates is encouraged, but you must write up assignments individually and in your own words. Please avoid using the internet as a source for solutions to problems, and do not under any circumstances use AI to generate your answers. Outsourcing your homework to AI will leave you unprepared for the exams and may open you up to a charge of academic dishonesty.
Homework is due by 5 PM on the due date and should be uploaded to Gradescope. For late homework, you will need to request permission for an extension. I am happy to be flexible in providing occasional short extensions when appropriate, but please contact me before the due date if at all possible.
Exams: There will be two 75-minute midterm exams and a final.
If you have two final examinations scheduled at the same time, it is the responsibility of the other department to provide an alternate exam.
Grading: The final course grade will be determined by:
Homework: 20%;
Midterm exams: 20% each;
Final exam: 40%.
Note: CourseWorks will attempt to create a running course grade. Please ignore this. Feel free to ask me if you have any questions concerning your course standing.
Help: My office hours are (tentatively) Tuesdays and Thursdays, 1--2 PM, and you should always feel free to email. Help is also available without appointment in the Mathematics Help Room whenever it is open.
Student Attendance and Participation: Regular attendance and participation are important for your success in this course. Attendance is not formally recorded, but students are expected to attend class, stay engaged, and keep up with assignments and announcements. While it may be tempting to think that it is possible to follow the course solely by reading the class notes and perhaps supplementary texts, almost all students benefit from the immediacy of lectures, the opportunity to ask questions, and the motor memory provided by taking notes.
If you miss a class due to illness or personal circumstances, it is your responsibility to review the material and catch up. If you expect to miss an assignment, quiz, or exam, notify your instructor as soon as possible. Absences due to religious observance should be communicated at the start of the term so that appropriate arrangements can be made.
For extended absences or difficulties that affect your ability to keep up across multiple courses, contact your advising dean. They can help coordinate support and communicate with instructors when needed.
Religious Observances: Columbia University respects the religious beliefs and practices of its students, faculty, and staff. In accordance with New York State law, no student may be penalized for absences related to religious observance.
Students who expect to miss class, an exam, or an assignment due to a religious holiday must notify the instructor at the beginning of the term so that appropriate arrangements can be made. Even when absences are excused, students are responsible for keeping up with course requirements. If a conflict arises, it is the student’s responsibility to work with the instructor in advance to determine how the work will be completed.
Disability Issues: Columbia and Barnard are committed to ensuring equal access for students with disabilities by providing accommodations and support services, and by promoting a learning environment that is inclusive and responsive to individual needs. Students who are not yet registered but who may benefit from disability-related accommodations are encouraged to contact their school’s disability services office early in the semester for a confidential consultation.
In order to receive disability-related academic accommodations for this course, students must first be registered with their school Disability Services (DS) office. Detailed information is available online for both the Columbia and Barnard registration processes. Refer to the appropriate website for information regarding deadlines, disability documentation requirements, and drop-in hours:
Columbia (CC, SEAS, GS): Columbia Disability Services (DS)
Barnard: Barnard Center for Accessibility Resources and Disability Services (CARDS).
Once registered, your approved accommodations will be communicated to instructors directly by the disability services office. Instructors do not require any additional documentation from students. If you are eligible for exam-related accommodations, please be aware that advance notice is required. Late requests are not guaranteed and depend on space and staffing availability. If you are unable to take an exam due to illness, notify both Disability Services or CARDS and your instructor as soon as possible. Make-up exams are granted at the instructor’s discretion and must align with the course’s established policies. Medical documentation is not required for instructors and does not guarantee a rescheduled exam. Students are encouraged to direct health-related concerns to their advising dean and to Disability Services or CARDS.
Academic Dishonesty: The vast majority of students do not cheat. Anyone who does so devalues the hard work of the rest of the class and creates a bad atmosphere for all. Anyone found to have cheated on an exam will receive a failing grade for the course and be subject to administrative discipline. If you are struggling with the material or have a problem about an upcoming exam, please discuss it with me instead of resorting to cheating.
As far as homework is concerned, you are encouraged to collaborate with other students, ask me or the TAs questions, or seek clarification from other texts, but all submitted work must reflect your own understanding and reasoning. If you do consult external resources such as books, websites, classmates, or AI tools, you must cite them appropriately. Using unauthorized assistance, directly copying from other students, an online source, or AI, or submitting work (including AI-generated content) that is not your own without acknowledgment violates University policy and may result in disciplinary action.
Important dates:
September 3: First day of class
September 12: Last day to add/drop via SSOL
October 6: Midterm exam 1
October 7: Drop date
November 3--4: Election break
November 5: Midterm exam 2
November 13: Course withdrawal and pass/fail deadline
November 26--28: Thanksgiving break
December 8: Last day of class, last day to choose pass/fail
December 15: Final exam (tentative)