Florian Pop: Math 6020-6030 (Graduate Algebra)

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

Grader for Math 6030: Zhenyue Guan

E-mail: zyguan AT sas.upenn.edu
Office/Phone: 3W1 DRLB

About this course:

• Required background/prerequisites: An axiomatic introduction to sets, maps, relations; the rigorous construction of the natural numbers, the integer number, the rational numbers. We will review this material during the beginning of the course, but (very) fast-paced. Basic facts about algebraic structures [monoids, groups, rings, (skew)fields, modules and vector spaces, etc.] would be useful, b/c otherwise the learning curve might be very steep.
• Description/Syllabus of the course: Math 6030 is the second part of the graduate algebra sequence Math 6020-6030.
  Topics (tentatively and in no specific order) for the sequence Math 6020-6030:
  • Groups: Basics (homomorphism, Noether isomorphism theorems, 5-lemma, snake lemma); Group actions, class equation, Sylow theorems; simple groups; composition series, Jordan-Holder theorem; Nilpotent and solvable groups; Semi-direct products and group extensions; Introduction to group cohomology; The category of groups: products, free products, free groups, inductive/projective limits of groups, quotient/completion functors (solvable, nilpotent; profinite, etc.).
  • Rings and (Skew)Fields. Basics: homomorphisms, left/right/two sided ideals, Noether isomorphism theorems; Prime and maximal ideals, nilradical and Jacobson radical; The category of rings, respectively of R-algebras. Rings of fractions, localization; Special classes of commutative rings: Euclidean rings, principal ideal domains, uniques factorization domains, Gauss lemma; Noetherian rings: basics, Hilbert basis theorem, primary decomposition, Krull dimension, Krulls Hauptidealsatz, Krull's height theorem, etc. Special classes of rings: Wedderburn (little) Thm, Artin-Zorn Thm, Wedderburn-Artin Thm, Jacobson Density Thm.
  • Modules and Vector spaces. Basics about the category of R-modules, respectively of F-vector spaces: Homomorphisms, Noether isomorphism theorems, exact sequences, 5-lemma, snake lemma; Generation of modules/vector spaces, basis and dimension; Nakayama lemma; Dual module/vector space, bilinear and quadratic forms, multilinear forms, symmetric/alternating forms; Determinant of an endomorphism, properties, characteristic polynomial, Cayley-Hamilton Thm; Tensor product, flatness, (symmetric/alternating) tensor algebra. Localization of modules, local properties; Free/projective/injective modules, injective hull; Projective/injective resolutions and the functors Tor and Ext; Noetherian/Artinian modules, normal series; etc.
    Linear algebra. Basics: Morphisms of free modules of finite rank and matrices; Bilinear symmetric/alternating forms and matrices; Eigenspaces, eigenvalues, minimal polynomial, primary decomposition, Jordan Canonical Form. Over the real/complex numbers: Inner product spaces (orthogonality, adjoints, self adjoint operators, isometries, normal operators, spectral theorem).
  • Fields and Galois Theory. Fields and field extensions: prime fields, algebraic and transcendental elements, finite extensions, algebraic extensions, transcendental extensions; Splitting fields and algebraic closure, normal extensions; Norm and trace, discriminant, separability: separable extensions, separable polynomials, perfect fields; separable degree, primitive element theorem, separable and perfect closure; Galois extensions: algebraic independence of homomorphisms, normal basis theorem, fixed fields, fundamental theorem of Galois theory, base change of Galois extensions; Galois group of a polynomial, respectively of generic polynomials; Cyclotomic extensions: roots of unity, cyclotomic fields. Solvability by radicals: abelian extensions, radical extensions; Hilbert's theorem 90, Kummer and Artin-Schreier Theory; Galois descent, fundamental theorem of algebra; Constructions with compass and straight edge, the regular n-gon; Infinite Galois theory. Transcendental extensions: linear disjointness, separating transcendence bases, separability and differentials.
  • More about commutative Rings. Integral ring extensions, finiteness, going up/down, integrally closed domains; Valuation rings: Chevalley's valuation theorem, characterization of integrally closed domains; Integral extensions of Noetherian rings and applications: Noether normalization theorem, Hilbert Nullstellensatz, algebraic sets; Dimension and transcendence degree; Regular local rings. Rings of dimension 1: characterization of discrete valuation rings, Dedekind domains, fractional ideals, Cartier divisors, ideal class group, projective modules over PIDs. Tentatively: Hilbert decomposition theory.
• Resources: There are several books you can use (and some course notes might be provided). You might check the source whose style is more suitable for you.
  • Algebra, by T. W. Hungerford
    
  • Algebra, by Serge Lang, GTM 2011, Springer Verlag.
    
  • Commutative Algebra of BOURBAKI...
    
  • Topics in Algebra, by Herstein (2nd Edition).
    
  • Algebra, by Galier and Schatz
    
  • Algebra I, II, by van der Waerden
    
  • Algebra I, II, by Jacobson

Basic Rules:

• It is recommended that you come regularly to the class and take notes, b/c we will not strictly follow any source/book.
• The final grade is based a midterm and a final exam (20%+30%) and everything else (50%). "Everything else" consists of regular homework, participation/performance in class, etc.
  • Exam dates (tentatively): March 14, May 2, 2024.
• 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 603. The homework assignment of each week is tentatively due on Friday of the next week (in my or grader's mailbox).
  • 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, preferably hand-written.
  • 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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