Florian Pop: Math 370 (Algebra I)

E-mail: pop AT math.upenn.edu
Office/Phone/Fax: DRL 4E7A / 215-898-5971 / 215-573-4063
Office hours: By appointment

Teaching Assistant: Yaojie Hu

E-mail: yaojie AT sas.upenn.edu
Office/Phone: TBA
Office hours: TBA

General Information

• See Undergrad Course Description
• Class: MW at 12:00-1:29 PM in TBA. First class on We, Aug 26, 2026
• Lab: T 5:15-6:45 PM, R 7:00-8:30 PM in TBA.
• This is the first part of a (rigorous proof based) introduction to abstract Algebra. For a passing grade, the students are expected to:
  • Understand and know the material and be able to solve related problems.
  • Understand rigorous mathematical argumentation/proofs and to be able to write coherent mathematical proofs.
• Syllabus:
  Basic facts about Sets, Maps, Relations. The natural numbers: Definition, Induction Principle, basic operations and properties, natural ordering, divisibility. The ring of integer numbers and field of rational numbers. Composition laws, basic algebraic structures: Monoids, Groups, Rings, (skew) Fields; Modules/Vector spaces. Products of algebraic structures. Groups: Basic group theory, e.g. (normal) Subgroups, quotient Groups; Morphisms, Norther Isomorphism Theorems. Examples. Possible supplements: Structure of finite abelian groups; Syllow Theorems. Rings: Basic facts, Examples (matrix rings, group rings, formal power series / polynomial rings, etc.); Subrings, Ideals, Ring homomorphisms, Quotient rings; Prime ideals, Maximal ideals, Comaximal ideals, Chinese Remainder Thm; Ring/Field of fractions. Examples. Possible supplements: Special classes of rings (Euclidean domains, Principal Ideal Domains, Unique factorization Domains, Noetherian Rings). Examples. Modules & Vector spaces: Freeness, Morphisms and Matrices, etc.
  
• Required background/prerequisites: Basic facts about sets and maps, familiarity with proofs, e.g proofs by induction, as covered in Math 202 and/or Math 203, see e.g. Math 370 Basics, Math 370 Test-HW. We will cover the Math 370 Basics material during the first two weeks or so, but fast-paced.
  
• Resources: There are several books you can use (and some course notes will be provided). You might check the source whose style is more suitable for you.
  • Algebra: Abstract and Concrete, Edition 2.5, by Frederick M. Goodman, especially Appendix A, B, C, D, E, sections of Ch 1--3, 6, 8.
    
  • Introduction to Abstract Algebra by Nicholson (4th Edition), especially sections form Ch 0--5, 8.
    
  • A first course in Abstract Algebra by Raleigh (7th edition), especially sections from Ch I--V, VII, IX.
    
  • Topics in Algebra by Herstein (2nd Edition), especially sections form Ch 1--3, 4.1, 4.5.
    
  • Algebra by Serge Lang, GTM 2011, Springer Verlag. This is a standard reference text which contains a large amount of material.

Basic Rules:

• The final grade is based on midterms and a final exam (10%+10%+30%) and everything else (50%). "Everything else" consists of regular homework, participation/performance in class, etc.
  • Exam dates (tentatively): Sept 30, Nov 2, Dec 2, 2026.
• Miscellannia: Announcements, homework assignment, notes, etc., will all be posted on web. No hard copies will be distributed. Please check this page frequently for the most updated information. Remember to use to RELOAD button of your browser.
• Homework:
  • Homework will be assigned each week, and in oder to see the homework please follow the links under Homework Math 370. The homework assignment of each week is tentatively due on Wednesday of the next week.
  • Your works should contain complete solutions, and rigorous and logically correct proofs for theoretical problems. (Note: such a proof must be written in grammatically correct language.)
  • You are encouraged to work in groups and discuss/communicate with each other as much as possible. But the work you hand in must be your own write-up.
  • Late work will not be accepted.

• Grading Note: At the end of the semester, everyone who has not withdrawn from the class will get a grade. The grade "I" (Incomplete) will not be given to avoid the grade "F" (Fail).

Info pages for undergraduate math:

• Mathematics Majors (information and advice for current and prospective).
• Biological Mathematics major
• Mathematics Minor requirements
• Actuarial Mathematics minor program

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