Teaching in summer semester 2026:
Retake exam Algebra
The retake exam will take place in room B 349 on Thursday, April 2nd, from 10:00 to 12:00 (start 10:10 ends 11:50). The students who want to take part at the retake exam should register by sending me an email and giving me their Matrikelnummer.
I aslo ask for those who can register through LSF to also register to the Retake Exam. It should possible a couple of days before it.
Algebra 2
This lecture will start with some complement on Galois theory, mostly to describe for P a polynomial with integer coefficients and L|Q its splitting field how the decomposition of P modulo a prime number p gives informations on the Galois group of L over Q. An application will be to prove that the Galois group over Q of a generic polynomial of degree n with integer coefficients is the symmetric group S_n. Then we will start the main topic of this lecture, commutative algebra. We will start by some generalities, will introduce the spectrum of a commutative ring, that is to say the set of its prime ideals endowed with the Zariski topology. We will introducing various properties: localisation, notherianity, dimension, etc... We will introduce the notion of Dedekind domain, and study its main properties. This rings are very important in number theory, as well as in algebraic geometry, in the sense that they are the correct notion of "smooth curves". Then we will move to study the category of R-modules over a commutative ring R: Exact sequences, Tensor products, localisation, flatness, etc... Then we will study more in details the theory of Krull dimension for finite type algebra over a field k, user Noether presentation Lemma, and some of its applications: Hilbert Nullstellensatz, the fact that the dimension of a finite type algebra A over a field k which is an integral domain is equal to the transcendence degree over k of its fraction field. Etc.. We will end by introducing and studying the notion of regular rings, and characterize the local regular rings.
This lecture is the sequel to the lecture "Algebra" I gave in the winter semester. Any student who wants to take this lecture should be aware of the content of the Algebra lecture.
Below you can find an updated version of the notes of Algebra taken and written by Leon Man:
here: Script (updated 23.03)
Lectures: Tuesday 10:00-12:00 (B132) and thursday 10:00-12:00 (B132). Exercise class Tuesday 16:00-18:00 (B132).
Tutorial classes (with Laurenz Wiesenberger): TBA.
Exam: on thursday 16th of July, 10:00-12:00 Room B132.
References:
S. Lang : Algebra, Springer (in english).
M. F. Atiyah, I. G. MacDonald: Introduction to Commutative Algebra. Addison-Wesley, Reading MA 1969
The lecture Alegbra 2 is ment to be a path, or an introcuction, to the lecture Algebraic Geometry of next year. I will give in Parallel a lecture on "Category theory" see below. It is recommended to take it, but not mandatory.
Category theory
This lecture will a complement to the lecture Algebra 2. It will be 2 hours each two weeks. It is an introcution to Category theory, and is ment to help to see/understand the lecture Algebra 2 (and also it sequel Algebraic Geometry) in a more global way. It is not mandatory and it is interesting in itself for any student who wants to study pure mathematics in the Master program.
If a student is interested in giving a talk at the end of the lecture, he is welcome and that could count as a seminar talk in the Bachelor program.
Lectures: Friday 10:00-12:00 (B134). Starts on Friday 17 April.
Reference: S. Mac-Lane, Categories for the working mathematician, Springer