Florian Pop: Math 371 (Algebra II)

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

Teaching Assistant: Dominick Villano

Office/Phone: DRL 4C13 / 617-833-2810
Office hours: Tue, 4:30-6:30 PM

General Information

• See Undergrad Course Description
• MWF at 10:00-11:00 AM in DRL 4C4. First class on We, Aug 29, 2019.
• This is the second part of a (rigorous proof based) introduction to 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:
  Rings: Basic facts, Examples (polynomial rings, matrix rings, group rings, etc.); Subrings, Ideals, Ring homomorphisms, Quotient rings; Prime ideals, Maximal ideals, Comaximal ideals, Chinese Remainder Thm; Rings of fractions. Special classes of rings: Euclidean domains, Principal Ideal Domains, Unique factorization Domains. More on rings of polynomials. Modules and Vector spaces: Basic facts, Examples; Homomorphisms, Isomorphism Thm. Multilinear Forms and Tensor Product. Finite modules over PID, Application: The Jordan Canonical Form. Fields: Basic facts, Examples, Finite Fields, Fields of fractions. Field Extensions, Algebraic (in)dependence, Algebraic extensions, Algebraic closure. Splitting fields, The fundamental Theorem of Galois Theory, Examples of Galois groups. Cyclotomic Fields, Straightedge & Compass constructions. Impossibility of resolving the quintic.
  
• Required background/prerequisites: Basic facts about sets and maps, familiarity with proofs, groups (as in Math 370), linear algebra (as in Math 314),
  
• Resources: There are several books you can use (and some course notes might/will be provided).
  • Abstract Algebra by Dummit & Foote (3rd Edition), Part II, Part III (most of this is linear algebra, which you should be familiar with), Part IV.
    
  • A first course in Abstract Algebra by Raleigh (7th edition), Ch: IV, V, VI, IX, X. Especially if you are not very familiar with proofs, you might find this text very useful/appealing.
    
  • Serge Lang, Algebra, Springer Verlag, standard reference text which contains a very large amount of material.

Basic Rules:

• The final grade is based on midterms and a final exam (30%+30%) and everything else (40%). "Everything else" consists of regular homework, participation/performance in class, etc.
  • Exam dates (tentatively): Oct 5, Nov 9, Dec 7, 2018.
• 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 371. 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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