Algebra and Number Systems
Modern Algebra Fall 2020
An Asynchronous Online Course
sormanic@gmail.com
MAT314: The syllabus of this course is continued in MAT 315. Algebraic structures such as groups, rings, and fields; their relations and applications to school and college mathematics, including the number systems of arithmetic and analysis. PREREQ: MAT 313.
MAT615: Mathematical induction. Introduction to group theory with examples-permutation groups, general linear group. Homomorphisms, subgroups, and congruences. Introduction to theory of rings and fields. Applications to number systems and number theory.
Expectations: Students are expected to learn both the mathematics covered in class and the mathematics in the textbook, as well as any other assigned reading. Completing homework is a significant part of the learning experience. Students should review topics from prior prerequisite courses as needed.
Online Asynchronous Lessons: Students should plan to complete two lessons per week which will appear linked below starting on August 26. For each lesson students will watch videos and do classwork between videos for 100 minutes. The solutions to the classwork will be included in the videos. Some students may choose to spend more time on the classwork before watching the solution. Some students may choose to complete a single lesson in two sittings.
Homework and Office Hours: Approximately three hours of homework will be assigned per lesson. It should be completed and submitted before starting the next lesson. All proofs should be written in numbered lines with statements and justifications in the style taught in class. All students will create a google doc following this template with their course number (MAT314 or MAT615) and their full name (for example MAT314-Sormani-Chris) and share it with the professor as an editor (sormanic@gmail.com). Photos of the homework and other questions will be uploaded into the document and the professor will answer your questions inside the document after being sent an email requesting that they look at it. Office hours are asynchronous. Questions may be asked any time of day and will be answered within 24 hours after the email is sent. Students may seek help and work together when doing homework. Always give credit for information learned online, with the help of a tutor, or a classmate. Simply cite where the help came from by providing a link, or a textbook, or the full name of the assistant.
Projects: Students taking MAT615 must complete a research project on number theory.
Examinations: There will be three 2-part 60-minute exams and an accumulative 3-part 90-minute final exam. Each part must be completed in 30 minutes and each student will have a slightly different exam. Students may consult books, notes, videos, and online resources, but may only take 30 minutes to complete a part of an exam. Students may not seek help from people or tutoring services during exams. Students will schedule the date and time of their exam with their professor sometime between Sunday 10pm and Monday 11pm. Exam 1 will be taken when a student has successfully completed the homework on Lessons 1-5. Exam 2 will be taken after completing Lessons 7-13. Exam 3 will be taken when students have successfully completed Lessons 15-20. The Final will be taken during Finals Week. There will be no retakes of exams.
Grading Policy: Each exam is worth 20%. The final is worth 30%. The homework/projects are worth 10%.
Materials, Resources and Accommodating Disabilities:
Textbook: Dan Saracino, Abstract Algebra, Second Edition, 2008. ISBN: 0-201-76390-7.
Resources: Each lecture will have a google doc and videos that can be viewed and reviewed as needed.
Games: Students should purchase a Rubik's Cube and Suduko is free online.
Inspiring Talk on Games by a Guest Speaker Dr. Manuel Luis Rivera
Additional Resources: A nice webpage about number theory useful for MAT615 students.
Accommodating Disabilities: Lehman College is committed to providing access to all programs and curricula to all students. Students with disabilities who may need accommodations are encouraged to register with the Office of Student Disability Services. For more information, please contact the Office of Student Disability Services, Shuster Hall, Room SH-238, telephone number, 718-960-8441. E-mail: disability.services@lehman.cuny.edu
Course Schedule:
(click on each lesson for the lesson googledoc and links to videos and photos of the lessons)
Lesson 1: Direct Proofs and Indirect Proofs about Sets
Lesson 2: Proof by Induction and the Well-Ordering Principle
Lesson 3: Binary Operations and Sets
Lesson 4: Groups and Matrices
Lesson 5: Finite Groups and the Division Algorithm
FAQ: Answers to Frequently Asked Questions
Lesson 6: Exam 1 (aim to take on Sept 13 or 14)(must take before October 20)
(proofs must be done in the style taught in class)
Part 1: A Proof by Induction (30 minutes)
Part 2: A Proof that Something is a Group (30 minutes)
Lesson 7: Theorems about Groups (Sept 15-16)
Lesson 8: A Talk by Alexander Diaz Lopez (Sept 21-22) (may skip if behind schedule)
Lesson 9: Powers of Elements, Double Induction, and Cyclic Groups (Sept 23-24)
Lesson 10: Subgroups (Sept 28-29)
Lesson 11: Direct Products (Oct 1-2)
Lesson 12: Functions (Oct 14-15)
Lesson 13: Symmetric Groups and Alternating Groups (Oct 18-19)
Lesson 14: Samples to Practice for Exam 2 (Oct 20-21)
Lesson 15: Exam 2 (aim to take on Oct 25-26) (must take before Nov 17)
(responses must be done in the style taught in class)
Part 1: Short answers about powers of elements, cyclic groups, and subgroups (30 minutes)
Part 2: Short answers about direct products and symmetric groups (30 minutes)
Lesson 16: Equivalence Relations and Equivalence Classes and Cosets (Oct 27-28)
Lesson 17: Cosets in Finite Groups and Lagrange’s Theorem (Nov 9-10)
(MAT313 students may skip Lesson 17 if far behind but cosets are on Exam 3)
Lesson 18: Group Homomorphisms (Nov 11-12)
Lesson 19: Group Isomorphisms (Nov 16-17)
Lesson 20: Fundamental Theorem of Finite Abelian Groups (Nov 18-19) (may skip if behind) also briefly surveys normal subgroups, quotients of groups, and Sylow’s Theorems
Lesson 21: Samples to Practice for Exam 3 (Nov 20-21)
Lesson 22: Exam 3 (aim to take on Nov 22-23) (must take before Dec 15)
The withdrawal deadline is Dec 13!
(responses must be done in the style taught in class with two column proofs)
Part 1: A Proof about Equivalence Relations/Classes and Cosets
Part 2: A Proof that a Given Function is a Group Homomorphism and/or Isomorphism
After Exam 3 there is no resubmission or feedback on the homework. You will check your work yourself and hand in perfect work. If you are unsure you may ask questions.
Lesson 23: Rings and Ideals (Nov 25-26)
Lesson 24: Inspiring Talk by Dr. Ashley Wheeler (Dec 2-3) (may skip if behind)
Lesson 25: Ring Homomorphisms and Isomorphisms (Dec 7-8)
Lesson 26: Fields (Dec 9-10) (MAT615 projects on Number Theory are due at the end of finals week)
Lesson 27: Polynomials over Rings (Dec 14-15)
Lesson 28: Polynomials over Fields, Long Division, and Roots (Dec 16-17)
Lessons 23-28 are on the final and should be submitted the day before you take the exam. Students who passed three exams may take an incomplete in the course.
Finals Week: Three Part Final Exam
(responses must be done in the style taught in class)
The Final will be given Sun Dec 20 at 9:30pm, Mon Dec 21 at 3 pm, and Mon Dec 21 at 9:30 pm.
Part I: A Proof by Contradiction that something is not a Group
Part 2: A Proof that a given function is a Ring Homomorphism and/or Isomorphism
Part 3: A Proof by Induction about Polynomials over Fields
Only students who have passed all three exams can take an incomplete and postpone the final.