Florian Pop: Teaching
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:
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